1 review in exponents and power equations

Review on Exponents

Let’s review the basics of exponential notation.Review on Exponents

base

exponentThe quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Let’s review the basics of exponential notation.Review on Exponents

N times

base

exponent

Rules of Exponents

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Let’s review the basics of exponential notation.Review on Exponents

N times

base

exponent

Multiply–Add Rule:

Rules of Exponents

Divide–Subtract Rule:

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule:

Let’s review the basics of exponential notation.Review on Exponents

N times

base

exponent

Multiply–Add Rule: ANAK = AN+K

Rules of Exponents

Divide–Subtract Rule:

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule:

Let’s review the basics of exponential notation.Review on Exponents

N times

base

exponent

Multiply–Add Rule: ANAK = AN+K

Rules of Exponents

Divide–Subtract Rule:AN

AK = AN – K

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule:

Let’s review the basics of exponential notation.Review on Exponents

N times

base

exponent

Multiply–Add Rule: ANAK = AN+K

Rules of Exponents

Divide–Subtract Rule:AN

AK = AN – K

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule: (AN)K = ANK

Let’s review the basics of exponential notation.

x9

Review on Exponents

N times

base

exponent

Multiply–Add Rule: ANAK = AN+K

Rules of Exponents

Divide–Subtract Rule:AN

AK = AN – K

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule: (AN)K = ANK

Let’s review the basics of exponential notation.

For example, x9x5 =x14 , x9

x5 = x9–5 = x4, and (x9)5 = x45.

Review on Exponents

N times

base

exponent

Multiply–Add Rule: ANAK =AN+K

Rules of Exponents

Divide–Subtract Rule:AN

AK = AN – K

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule: (AN)K = ANK

Let’s review the basics of exponential notation.

For example, x9x5 =x14 , x9

x5 = x9–5 = x4, and (x9)5 = x45.

Review on Exponents

N times

These particular operation–conversion rules appear often in other forms in mathematics.

base

exponent

Multiply–Add Rule: ANAK =AN+K

Rules of Exponents

Divide–Subtract Rule:AN

AK = AN – K

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule: (AN)K = ANK

Let’s review the basics of exponential notation.

For example, x9x5 =x14 , x9

x5 = x9–5 = x4, and (x9)5 = x45.

Review on Exponents

N times

These particular operation–conversion rules appear often in other forms in mathematics. Hence their names, the Multiply–Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule, are important.

base

exponent

Multiply–Add Rule: ANAK =AN+K

Rules of Exponents

Divide–Subtract Rule:AN

AK = AN – K

The quantity A multiplied to itself N times is written as AN.

A x A x A ….x A = AN

Power–Multiply Rule: (AN)K = ANK

Let’s review the basics of exponential notation.

For example, x9x5 =x14 , x9

x5 = x9–5 = x4, and (x9)5 = x45.

Review on Exponents

N times

These particular operation–conversion rules appear often in other forms in mathematics. Hence their names, the Multiply–Add Rule, the Divide–Subtract Rule, the Power–Multiply Rule, are important. Let’s extend the definition to negative and fractional exponents.

Since = 1A1

A1

The Exponential Functions

Since = 1 = A1 – 1 = A0A1

A1

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since =1AK

A0

AK

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, 1AK

A0

AK

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Since (A )k = A = (A1/k )k, k

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

The Exponential Functions

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Fractional Powers: A1/k = A. k

The Exponential Functions

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by of the exponent last.

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify.

b. 91/2 =

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 =

c. 9 –3/2 =

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify. 1 92

1 81

b. 91/2 =

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 = =

c. 9 –3/2 =

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify. 1 92

1 81

b. 91/2 = √9 = 3

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 = =

c. 9 –3/2 =

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify. 1 92

1 81

b. 91/2 = √9 = 3

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 = =

c. 9 –3/2 =

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify. 1 92

1 81

b. 91/2 = √9 = 3

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 = =

c. 9 –3/2 = (9½)–3

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Pull the numerator outside to take the root and simplify the base first.

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify. 1 92

1 81

b. 91/2 = √9 = 3

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 = =

c. 9 –3/2 = (9½)–3 = 3–3

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Pull the numerator outside to take the root and simplify the base first.

Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1

A1

0-Power Rule: A0 = 1

Since = = A0 – K = A–K, we get the Negative Power Rule.1AK

A0

AK

Negative Power Rule: A–K = 1AK

Example A. Simplify. 1 92

1 81

1 33

b. 91/2 = √9 = 3

a.

Fractional Powers: A1/k = A. k

The Exponential Functions

For a general fractional exponent, we interpret the operationsstep by step by doing the numerator of the exponent last.

9–2 = =

c. 9 –3/2 = (9½)–3 = 3–3 = = 1 27

Since (A )k = A = (A1/k )k, hence A1/k = A. k k

Pull the numerator outside to take the root and simplify the base first.

The operation of taking power may be passed to factors.Fractional Exponents

The operation of taking power may be passed to factors.Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN(AB)N =ANBN and

The operation of taking power may be passed to factors.Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36.

(AB)N =ANBN and

The operation of taking power may be passed to factors.Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

The operation of taking power may be passed to factors.Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

The operation of taking power may be passed to factors.Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

x–1/2y2/3

Example B. Simplify the exponents.

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

x–1/2y2/3

Example B. Simplify the exponents.

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

Do not covert the exponents into radicals. We are to collect them using arithmetic.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

=x*x2/3y3

x–1/2y2/3

2*1/3 2*3/2

Example B. Simplify the exponents.

x–1/2y2/3

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

=x*x2/3y3

x–1/2y2/3=

2*1/3 2*3/2

x5/3y3

1+2/3

Example B. Simplify the exponents.

x–1/2y2/3 x–1/2y2/3

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

=x*x2/3y3

x–1/2y2/3=

2*1/3 2*3/2

=

x5/3y3

1+2/3

x5/3 – (–1/2) y3 – 2/3

Example B. Simplify the exponents.

x–1/2y2/3 x–1/2y2/3

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

=x*x2/3y3

x–1/2y2/3=

2*1/3 2*3/2

=

x5/3y3

1+2/3

x5/3 – (–1/2) y3 – 2/3

Example B. Simplify the exponents.

x–1/2y2/3 x–1/2y2/3

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

The operation of taking power may be passed to factors.

x(x1/3y3/2)2

=x*x2/3y3

x–1/2y2/3=

2*1/3 2*3/2

=

x5/3y3

1+2/3

x5/3 – (–1/2) y3 – 2/3

= x13/6 y7/3

Example B. Simplify the exponents.

x–1/2y2/3 x–1/2y2/3

= x5/3 +1/2 y7/3

Fractional Exponents

The Power–Passing Rules:

( ) AB =

N AN

BN

For example, (2*3)2 = 22*32 = 36. However the operation of taking power does not pass to terms.

(AB)N =ANBN and

For example, (2 + 3)2 ≠ 22 + 32.

In the example below, we use these rules to collect exponents.

Review on Exponents and Power Equations Power Equations

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

The reciprocal of the power 3

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

The reciprocal of the power 3

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

Rational Power equations are equations of the type xR = c where R = P/Q is a rational number.

The reciprocal of the power 3

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

Rational Power equations are equations of the type xR = c where R = P/Q is a rational number. To solve them, we take the reciprocal power,

The reciprocal of the power 3

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

Rational Power equations are equations of the type xR = c where R = P/Q is a rational number. To solve them, we take the reciprocal power, that is,if xR = c,

The reciprocal of the power 3

or xP/Q = c

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

Rational Power equations are equations of the type xR = c where R = P/Q is a rational number. To solve them, we take the reciprocal power, that is,if xR = c,

then x = (±)c1/R

Reciprocate the powers

The reciprocal of the power 3

or xP/Q = c

x = (±)cQ/Por

Review on Exponents and Power Equations Power EquationsThe solution to the equation x 3 = –8 is x = √–8 = –2.

3

Using fractional exponent, we write these steps asif x3 = –8 then

x = (–8)1/3 = –2.

Rational Power equations are equations of the type xR = c where R = P/Q is a rational number. To solve them, we take the reciprocal power, that is,if xR = c,

then x = (±)c1/R

Reciprocate the powers

The reciprocal of the power 3

However, depending on the values of c and Q/P, it may be that there is no real solutions, exactly one real solutions, or both (±) c1/R are real solutions.

or xP/Q = c

x = (±)cQ/Por

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3x = √64 = 4.

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

x = √64 = 8.

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

c. x2 = –64

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

c. x2 = –64 x = (–64)1/2 so there is no real number solution.

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

c. x2 = –64 x = (–64)1/2 so there is no real number solution.

d. x –2/3 = 64

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

c. x2 = –64 x = (–64)1/2 so there is no real number solution.

d. x –2/3 = 64 x = 64–3/2

x = (√64)–3

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

c. x2 = –64 x = (–64)1/2 so there is no real number solution.

d. x –2/3 = 64 x = 64–3/2

x = (√64)–3 = 4–3 = 1/64.

Review on Exponents and Power Equations Example C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that

3

We note that this is the only solution.x = √64 = 4.

b. x2 = 64 x = 641/2 or that

We note that both ±8 are solutions. x = √64 = 8.

c. x2 = –64 x = (–64)1/2 so there is no real number solution.

d. x –2/3 = 64 x = 64–3/2

x = (√64)–3 = 4–3 = 1/64.

Again we check that both ±1/64 are solutions.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

a. 2x2/3 – 7 = 1Example D. Solve for x.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8a. 2x2/3 – 7 = 1Example D. Solve for x.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4

a. 2x2/3 – 7 = 1Example D. Solve for x.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2

a. 2x2/3 – 7 = 1Example D. Solve for x.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1 = 6

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1 = 6

(2x + 1)1/3 = 2

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1 = 6

(2x + 1)1/3 = 22x + 1 = 23

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1 = 6

(2x + 1)1/3 = 22x + 1 = 23

2x = 23 – 1 = 7

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1 = 6

(2x + 1)1/3 = 22x + 1 = 23

2x = 23 – 1 = 7

x = 7/2

Review on Exponents and Power Equations For linear form of the power equations, we first isolate the term with the power, then apply the reciprocal power to solve for x.

2x2/3 = 8x2/3 = 4x = 43/2 = (41/2)3

a. 2x2/3 – 7 = 1Example D. Solve for x.

x = 8b. 1 = 7 – 3(2x + 1)1/3

3(2x + 1)1/3 = 7 – 1 = 6

(2x + 1)1/3 = 22x + 1 = 23

2x = 23 – 1 = 7

x = 7/2

We need calculators for irrational solutions which is our next topic.

Exercise A. Write the following arithmetic expressions in radicals and simplify.

1. 41/2 4. 491/23. 361/22. 251/2

9. (100x)1/2

925( )1/25. 6. (–8)1/3 8. 64( )1/3

10. 16–1/2 11. 64–1/2

7. 1251/3

x4( ) –1/212.

925( ) –1/214.13. –1

64( ) –1/3 x(15. )–1/2100

16. 163/2 17. 253/2 18. –164( ) –2/3 19. 125–2/3

Review on Exponents and Power Equations

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Exercise B. Combine exponents to simplify the expressions. Write the results in exponents and in radicals.

20. x1/3x4/3x1/2

24. x–1/4 x1/2

21. x3/4y1/2x3/2y4/3

23. y–2/3x–1/4y–3/2x–2/3

22. y–1/4x5/6x4/3y1/2

25. x1/4 x–1/2

27. y1/2x1/3

x4/3y–3/226. x–1/4

x–1/228. y–1/2x1/3

y–4/3x–3/2

29. (100x2)–1/2

(64x)1/3 30. (36x3)–1/2

(64x1/2)–1/3

Review on Exponents and Power Equations

Exercise C. Solve the following questions

31. x1/3 = 2 33. y–2/3 = 4 32. 2y –1/2 = 3

34. 3x1/3 + 1 = 2

37. y–2/3 – 5 = –1

35. 2 – 3y 1/2 = 5 36. 2 – 3y–1/2 = 5

38. 4 = 4y–2 – 5 39. 3x–2/3 – 10 = 2

40. (y – 2)–1/3 – 5 = 4 41. (4y + 3)–2 – 6 = 19

42. (3 – x)–3 = 8 43. (–2y + 5)3/2 – 6 = 21

44. 9(3 + 2x)–2/3 –1 = 3 45. –(–2x + 3)–3/2 – 6 = 2