Example:

• $3^2$
• Index $2$

Example:

• $27^{\frac{2}{3}} = 3^2 = 9$
• $16^{\frac{5}{4}} = 2^5 = 32$

Example:

Example:

$(0.16)^{\frac{3}{2}} = \left(\frac{16}{100}\right)^{-\frac{3}{2}} = \left(\frac{100}{16}\right)^{\frac{1}{2}} = \left(\frac{10}{4}\right)^3$

$= \left(\frac{5}{2}\right)^3 = 125/8$

$\begin{aligned} a^x \times a^y & = a^{x+y} \\ a^x \div a^y & = a^{x-y} \\ (a^x)^y & = a^{xy} \end{aligned}$

Example:

$(4x^2y)^{3} \times x^{-2}y \\= 64x^6y^3 \times x^{-2}y = 64x^4y^4$

Q1.

1. Evaluate $(0.2)^{-2}$:

$\left(\frac{1}{5}\right)^{-2} = 5^2 = :success[25]$

2. Simplify $\left(16a^{12}\right)^{\frac{3}{4}}$: $\left(16\right)^{\frac{3}{4}} = 2^3 = 8$

$:success[8a^9]$

Q2.

Find the value of each of the following:

3. $\left(\frac{5}{3}\right)^{-2}$:

$\left(\frac{5}{3}\right)^{-2}= \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = :success[\frac{9}{25}]$

4. $81^{\frac{3}{4}}$:

$81^{\frac{3}{4}} = 3^3 = :success[27]$

Q4.

1. Evaluate $9^{-\frac{1}{2}}$:

$9^{-\frac{1}{2}} = 3^{-1} = :success[\frac{1}{3}]$

2. Simplify $\frac{(4x^4)^3 y^2}{2x^2 y^5}$:

$\frac{64x^{12} y^2}{2x^2 y^5} = :success[32x^{10} y^{-3}]$

Summary

• Multiplying with same base: Add exponents
• Dividing with same base: Subtract exponents
• Raising a power to another power: Multiply exponents
• Product raised to a power: Distribute power to each factor
• Quotient raised to a power: Apply power to numerator and denominator
• Zero exponent: Any non-zero base raised to the zero power is $:success[1]$
• Negative exponent: Reciprocal of the base raised to the positive exponent
• Fractional exponent: Numerator is the power, denominator is the root