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Evaluating Exponential Expressions

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Evaluating Exponential Expressions

What if you had an exponential expression requiring multiple operations, like 2\left(\frac{1}{4}\right)^2 - \left(\frac{1}{4}\right)^3 ? How could you simplify it? After completing this Concept, you'll be able to use the order of operations to evaluate exponential expressions like this one.

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Foundation: 02807S Evaluating Exponential Expressions

Guidance

When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS:

  1. Evaluate inside the Parentheses .
  2. Evaluate E xponents.
  3. Perform M ultiplication and D ivision operations from left to right.
  4. Perform A ddition and S ubtraction operations from left to right.

Example A

Evaluate the following expressions.

a) 5^0

b) \left(\frac{2}{3}\right)^3

c) 16^{\frac{1}{2}}

d) 8^{-\frac{1}{3}}

Solution

a) 5^0=1 A number raised to the power 0 is always 1.

b) \left(\frac{2}{3}\right)^3=\frac{2^3}{3^3}=\frac{8}{27}

c) 16^{\frac{1}{2}}=\sqrt{16}=4 Remember that an exponent of \frac{1}{2} means taking the square root.

d) 8^{-\frac{1}{3}}=\frac{1}{8^{\frac{1}{3}}}=\frac{1}{\sqrt[3]{8}}=\frac{1}{2} Remember that an exponent of \frac{1}{3} means taking the cube root.

Example B

Evaluate the following expressions.

a) 3 \cdot 5^2-10 \cdot 5+1

b) \frac{2 \cdot 4^2-3 \cdot 5^2}{3^2-2^2}

c) \left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}

Solution

a) Evaluate the exponent: 3 \cdot 5^2 - 10 \cdot 5+1=3 \cdot 25-10 \cdot 5+1

Perform multiplications from left to right: 3 \cdot 25-10 \cdot 5+1=75-50+1

Perform additions and subtractions from left to right: 75-50+1=26

b) Treat the expressions in the numerator and denominator of the fraction like they are in parentheses: \frac{(2 \cdot 4^2-3 \cdot 5^2)}{(3^2-2^2)}=\frac{(2 \cdot 16-3 \cdot 25)}{(9-4)}=\frac{(32-75)}{5}=\frac{-43}{5}

c) \left(\frac{3^3}{2^2}\right)^{-2} \cdot \frac{3}{4}=\left(\frac{2^2}{3^3}\right)^2 \cdot \frac{3}{4}=\frac{2^4}{3^6} \cdot \frac{3}{4}=\frac{2^4}{3^6} \cdot \frac{3}{2^2}=\frac{2^2}{3^5}=\frac{4}{243}

Example C

Evaluate the following expressions for x = 2, y = - 1, z = 3 .

a) 2x^2-3y^3+4z

b) (x^2-y^2)^2

c) \left(\frac{3x^2y^5}{4z}\right)^{-2}

Solution

a) 2x^2-3y^3+4z&=2 \cdot 2^2-3 \cdot (-1)^3+4 \cdot 3\\ &=2 \cdot 4-3 \cdot (-1)+4 \cdot 3=8+3+12\\ &=23

b) (x^2-y^2)^2=(2^2 - (-1)^2)^2=(4-1)^2=3^2=9

c)

\left( \frac{3x^2y^5}{4z}\right)^{-2}&=\left( \frac{3 \cdot 2^2 \cdot (-1)^5}{4 \cdot 3}\right)^{-2}\\ &=\left(\frac{3 \cdot 4 \cdot (-1)}{12}\right)^{-2}\\ &=\left(\frac{-12}{12}\right)^{-2}\\ &=\left(\frac{-1}{1}\right)^{-2}\\ &=\left(\frac{1}{-1}\right)^2\\ &=(-1)^2\\ &=1

Watch this video for help with the Examples above.

CK-12 Foundation: Evaluating Exponential Expressions

Vocabulary

  • When evaluating expressions we must keep in mind the order of operations. You must remember PEMDAS:
  1. Evaluate inside the Parentheses .
  2. Evaluate E xponents.
  3. Perform M ultiplication and D ivision operations from left to right.
  4. Perform A ddition and S ubtraction operations from left to right.

Guided Practice

Evaluate the following expression for x = 3, y = -2, z = -1 .

2z((x+1)^\frac{1}{2}-y^3)^2

Solution:

2z((x+1)^\frac{1}{2}-y^3)^2&=2(-1)(((3)+1)^\frac{1}{2}-(-2)^3)^2\\ &=-2(4^\frac{1}{2}+8)^2\\ &=-2(2+8)^2\\ &=-2(10)^2\\ &=-200

Practice

Evaluate the following expressions to a single number.

  1. 3^{-2}
  2. {-4}^{-3}
  3. (6.2)^0
  4. 8^{-4} \cdot 8^6
  5. \left (16^\frac{1}{2} \right )^3
  6. x^2 \cdot 4x^3 \cdot y^4 \cdot 4y^2 , if x = 2 and y = -1
  7. a^4(b^2)^3 + 2ab , if a = -2 and b = 1
  8. 5x^2 - 2y^3 + 3z , if x = 3, y = 2, and z = 4
  9. \left ( \frac{a^2}{b^3} \right )^{-2} , if a = 5 and b = 3
  10. \left ( \frac{x^{-2}}{y^4} \right )^\frac{1}{2} , if x=-3 and y=2

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