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# Evaluating Exponential Expressions

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Practice 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.

### 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.

### 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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