Radical & Exponential Functions - ACT Math

1. $2$ raised to the $4$th power equals $16$.

$a^n b^n$. Distribute the exponent to each factor in the product.

$3^8$. Use power rule: $(3^2)^4 = 3^{2 \cdot 4} = 3^8$.

$\frac{1}{16}$. Negative exponent means reciprocal: $\frac{1}{4^2} = \frac{1}{16}$.

1. $\sqrt[4]{16} = 2$ since $2^4 = 16$.

1. Any non-zero number raised to the power $0$ equals $1$.

$a^{m-n}$. Subtract exponents when dividing powers with the same base.

$a^{m+n}$. Add exponents when multiplying powers with the same base.

$4(\sqrt{2}-1)$. Multiply by conjugate $\frac{\sqrt{2}-1}{\sqrt{2}-1}$ to rationalize.

1. The number that when raised to the 4th power equals 81.

$x$. Add exponents when multiplying: $\frac{2}{3} + \frac{1}{3} = 1$, so $x^1 = x$.

$a>1$. Base greater than 1 creates exponential growth.

$x=\log_a(y)$. Logarithm is the inverse operation of exponentiation.

$(0,1)$. When $x=0$, any positive base equals 1.

$-3\le x\le 3$. Need $9-x^2 \ge 0$, so $x^2 \le 9$.

1. The exponent is the power to which the base is raised.

1. $2$ cubed means $2 \times 2 \times 2 = 8$.

$|x|$. Absolute value needed since $x$ can be negative.

$\frac{7}{8}$. $\sqrt{\frac{49}{64}} = \frac{\sqrt{49}}{\sqrt{64}} = \frac{7}{8}$.

1. $2$ multiplied by itself $5$ times equals $32$.

7 or -7. Taking the square root of both sides gives $x = \pm 7$.

$9$. $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$.

$3^8$. Use power rule: $(3^2)^4 = 3^{2 \cdot 4} = 3^8$.

$5^5$. Use quotient rule: $5^{7-2} = 5^5$.

$2^8$. Use product rule: $2^{3+5} = 2^8$.

1. $\sqrt{36} = 6$ since $6^2 = 36$.

$10^3$. $1000 = 10 \times 10 \times 10 = 10^3$.

1. $\sqrt{100} = 10$ since $10^2 = 100$.

1. $\frac{\sqrt{16}}{\sqrt{4}} = \frac{4}{2} = 2$.