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Exponential Terms Raised to an Exponent

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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 .

Vocabulary

Base
In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression 2^5 , ‘2’ is the base. In the expression (-3y)^4 , ‘ -3y ’ is the base.
Exponent
In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
In the expression 2^5 , ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: 2^5=2 \times 2 \times 2 \times 2 \times 2 .
In the expression (-3y)^4 , ‘4’ is the exponent. It means to multiply -3y times itself 4 times as shown here: (-3y)^4=-3y \times -3y \times -3y \times -3y .
Laws of Exponents
The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.

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}}

Practice

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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