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Exponential Terms Raised to an Exponent

Multiply to raise exponents to other exponents

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Power Rule for Exponents

Can you simplify an expression where an exponent has an exponent? For example, how would you simplify [(2^3)^2]^4 ?

Watch This

James Sousa: Properties of Exponents

Guidance

In the expression x^3 , the x is called the base and the 3 is called the exponent . Exponents are often referred to as powers . When an exponent is a positive whole number, it tells you how many times to multiply the base by itself. For example:

  • x^3=x\cdot x \cdot x
  • 2^4=2\cdot 2 \cdot 2 \cdot 2=16 .

There are many rules that have to do with exponents (often called the Laws of Exponents ) that are helpful to know so that you can work with expressions and equations that involve exponents more easily. Here you will learn a rule that has to do with raising a power to another power.

RULE: To raise a power to a new power, multiply the exponents.

& (a^m)^n = \underleftrightarrow{(a \times a \times \ldots \times a)^n}\\& \qquad \qquad \qquad \quad {\color{red}\downarrow}\\& \qquad \qquad \quad {\color{red}m} \ \text{{\color{red} factors}}\\& (a^m)^n=\underleftrightarrow{(a \times a \times \ldots \times a)} \times \underleftrightarrow{(a \times a \times \ldots \times a)} \ \underleftrightarrow{(a \times a \times \ldots \times a)}\\& \qquad \qquad \qquad \quad \ {\color{red}\downarrow} \qquad \qquad \qquad \qquad {\color{red}\downarrow} \qquad \qquad \qquad \quad {\color{red}\downarrow}\\& \qquad \quad \quad \underleftrightarrow{\quad {\color{red}m} \ \text{{\color{red}factors}} \qquad \qquad \ \ {\color{red}m} \ \text{{\color{red}factors}} \qquad \quad \ {\color{red}m} \ \text{{\color{red}factors}} \ \ }\\& \qquad \qquad \qquad \qquad \qquad \qquad \qquad {\color{red}n \ times}\\& (a^m)^n=\underleftrightarrow{a \times a \times a \ldots \times a}\\& \qquad \qquad {\color{red}mn \ \text{factors}}\\& (a^m)^n = a^{\color{red}mn}

Example A

Evaluate (2^3)^2 .

Solution: (2^3)^2=2^6=64 .

Example B

Simplify (x^7)^4 .

Solution: (x^7)^4=x^{28} .

Example C

Evaluate (3^2)^3 .

Solution: (3^2)^3=3^6=729 .

Example D

Simplify (x^2y^4)^2\cdot(xy^4)^3 .

Solution: (x^2y^4)^2\cdot(xy^4)^3=x^4y^8\cdot x^3y^{12}=x^7y^{20} .

Concept Problem Revisited

[(2^3)^2]^4=[2^6]^4=2^{24} . Notice that the power rule applies even when a number has been raised to more than one power. The overall exponent is 24 which is 3\cdot 2 \cdot 4 .

Guided Practice

You know you can rewrite 2^4 as 2 \times 2 \times 2 \times 2 and then calculate in order to find that \boxed{2^4=16} . This concept can also be reversed. To write 32 as a power of 2, 32=2 \times 2  \times 2  \times 2 \times 2 . There are 5 twos; therefore, \boxed{32=2^{\color{red}5}} . Use this idea to complete the following problems.

1. Write 81 as a power of 3.

2. Write (9)^3 as a power of 3.

3. Write (4^3)^2 as a power of 2.

Answers:

1. 81={\color{red}3} \times {\color{red}3}=9 \times {\color{red}3}=27 \times {\color{red}3}=81

There are 4 threes. Therefore \boxed{81=3^{\color{red}4}}

2. 9={\color{red}3} \times {\color{red}3}=9

There are 2 threes. Therefore \boxed{9=3^{\color{red}2}} .
(3^2)^3 Apply the law of exponents for power to a power-multiply the exponents.
3^{2 \times 3}=3^{\color{red}6}
Therefore \boxed{(9)^3=3^{\color{red}6}}

3. 4={\color{red}2} \times {\color{red}2}=4

There are 2 twos. Therefore \boxed{4=2^{\color{red}2}}
\left((2^2)^3\right)^2 Apply the law of exponents for power to a power-multiply the exponents.
\boxed{2^{2 \times 3}=2^{\color{red}6}}
(2^6)^2 Apply the law of exponents for power to a power-multiply the exponents.
\boxed{2^{6 \times 2}=2^{\color{red}12}}
Therefore \boxed{(4^3)^2=2^{\color{red}12}}

Explore More

Simplify each of the following expressions.

  1. \left(\frac{x^4}{y^3}\right)^5
  2. \frac{(5x^2y^4)^5}{(5xy^2)^3}
  3. \frac{x^8y^9}{(x^2y)^3}
  4. (x^2y^4)^3
  5. (3x^2)^2\cdot(4xy^4)^2
  6. (2x^3y^5)(5x^2y)^3
  7. (x^4y^6z^2)^2(3xyz)^3
  8. \left(\frac{x^2}{2y^3}\right)^4
  9. \frac{(4xy^3)^4}{(2xy^2)^3}
  10. True or false: (x^2+y^3)^2=x^4+y^6
  11. True or false: (x^2y^3)^2=x^4y^6
  12. Write 64 as a power of 4.
  13. Write (16)^3 as a power of 2.
  14. Write (9^4)^2 as a power of 3.
  15. Write (81)^2 as a power of 3.
  16. Write (25^3)^4 as a power of 5.

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