ACT Math Flashcards: Radical And Exponential Functions

What this deck covers

This deck focuses on Radical And Exponential Functions, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Find $x$ if $x^3 = 8$.

Answer:

1. Taking the cube root of both sides gives $x = 2$.

Flashcard 2: What is the exponent rule for a product raised to a power, $(ab)^n$?

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

Flashcard 3: What is $3^4$?

Answer:

1. $3$ raised to the $4$th power equals $81$.

Flashcard 4: What is the value of $3^{-2}$?

Answer: $\frac{1}{9}$. Negative exponents mean reciprocal: $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Flashcard 5: What is the base in an exponential function $y = ab^x$?

Answer: $b$. The base determines the growth or decay factor.

Flashcard 6: State the formula for exponential decay.

Answer: $y = a(1 - r)^t$. Where $a$ is initial value, $r$ is decay rate, $t$ is time.

Flashcard 7: Express $\text{√}(x^2)$ in simplest form.

Answer: $|x|$. Square root of $x^2$ is the absolute value of $x$.

Flashcard 8: Find $x$ if $x^2 = 49$.

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

Flashcard 9: Simplify $\text{√}16$.

Answer:

1. The positive number that when squared equals 16.

Flashcard 10: Express $\text{√}(x^2)$ in simplest form.

Answer: $|x|$. Square root of $x^2$ is the absolute value of $x$.

Flashcard 11: What is the exponent in $7^2$?

Answer:

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

Flashcard 12: Evaluate $(-2)^3$.

Answer: -8. Cube the negative number: $(-2) \times (-2) \times (-2)$.

Flashcard 13: Identify the domain of $f(x)=\sqrt{x-5}$.

Answer: $x\ge 5$. Expression under square root must be non-negative: $x-5 \ge 0$.

Flashcard 14: What is the definition of a zero exponent for $a \ne 0$, $a^0$?

Answer: $1$. Any nonzero number raised to the power of zero equals 1.

Flashcard 15: Simplify: $\text{square root of } 121$.

Answer:

1. $\sqrt{121} = 11$ since $11^2 = 121$.

Flashcard 16: Identify the simplified value of $(3^2)^4$.

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

Flashcard 17: What is the fourth root of 16?

Answer:

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

Flashcard 18: What is the square root of 49?

Answer:

1. $\sqrt{49} = 7$ since $7^2 = 49$.

Flashcard 19: What is the domain restriction for $f(x)=\sqrt{g(x)}$ over the reals?

Answer: $g(x)\ge 0$. Expression under square root must be non-negative.

Flashcard 20: Identify the simplified value of $\frac{5^7}{5^2}$.

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

Flashcard 21: What is the simplification of $\sqrt{\frac{a}{b}}$ for $a \ge 0$ and $b>0$?

Answer: $\frac{\sqrt{a}}{\sqrt{b}}$. Square root of quotient equals quotient of square roots.

Flashcard 22: What is the horizontal asymptote of $f(x)=a^x$ for $a>0$ and $a\ne 1$?

Answer: $y=0$. Exponential functions approach zero but never reach it.

Flashcard 23: What is the meaning of $a^{\frac{m}{n}}$ for $a \ge 0$ and integers $m,n>0$?

Answer: $\sqrt[n]{a^m}$. Take the $n$th root of $a$ raised to the $m$th power.

Flashcard 24: Identify the base in $5^3$.

Answer:

1. The base is the number being raised to a power.

Flashcard 25: State the formula for exponential decay.

Answer: $y = a(1 - r)^t$. Where $a$ is initial value, $r$ is decay rate, $t$ is time.

Flashcard 26: Simplify: $\frac{\text{square root of } 16}{\text{square root of } 4}$.

Answer:

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

Flashcard 27: Identify the simplified value of $\sqrt{72}$ in simplest radical form.

Answer: $6\sqrt{2}$. $\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$.

Flashcard 28: What is the solution form for $a^x=a^k$ when $a>0$ and $a\ne 1$?

Answer: $x=k$. Equal bases means equal exponents for exponential equations.

Flashcard 29: Identify the simplified value of $\sqrt{18x^2}$ for real $x$.

Answer: $3\sqrt{2}\,|x|$. $\sqrt{18x^2} = \sqrt{9 \cdot 2 \cdot x^2} = 3|x|\sqrt{2}$.

Flashcard 30: What is the value of $\text{√}1$?

Answer:

1. The square root of 1 is always 1.

Flashcard 31: State the value of $4^0$.

Answer:

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

Flashcard 32: Simplify: $\frac{\text{square root of } 16}{\text{square root of } 4}$.

Answer:

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

Flashcard 33: What is the value of $2^{-3}$?

Answer: $\frac{1}{8}$. Negative exponent means reciprocal: $\frac{1}{2^3} = \frac{1}{8}$.

Flashcard 34: Simplify: $\frac{\text{square root of } 64}{\text{square root of } 16}$.

Answer:

1. $\frac{\sqrt{64}}{\sqrt{16}} = \frac{8}{4} = 2$.

Flashcard 35: What is the definition of a rational exponent, $a^{\frac{1}{n}}$ for $a \ge 0$?

Answer: $\sqrt[n]{a}$. Fractional exponent $\frac{1}{n}$ equals the $n$th root.

Flashcard 36: What is the square root of 81?

Answer:

1. $\sqrt{81} = 9$ since $9 \times 9 = 81$.

Flashcard 37: What is the fourth root of 81?

Answer:

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

Flashcard 38: Identify the domain of $f(x)=\sqrt{\frac{x+1}{x-2}}$.

Answer: $(-\infty,-1]\cup(2,\infty)$. Need $\frac{x+1}{x-2} \ge 0$ with $x \ne 2$.

Flashcard 39: Identify the simplified value of $2^3\cdot 2^5$.

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

Flashcard 40: What is the value of $3^{-2}$?

Answer: $\frac{1}{9}$. Negative exponents mean reciprocal: $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$.

Flashcard 41: What is the definition of a rational exponent, $a^{\frac{1}{n}}$ for $a \ge 0$?

Answer: $\sqrt[n]{a}$. Fractional exponent $\frac{1}{n}$ equals the $n$th root.

Flashcard 42: State the value of $2^5$.

Answer:

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

Flashcard 43: What is the exponent rule for a quotient raised to a power, $\left(\frac{a}{b}\right)^n$?

Answer: $\frac{a^n}{b^n}$. Distribute the exponent to numerator and denominator separately.

Flashcard 44: What is $1$ as a power of any nonzero number?

Answer: $a^0$. Zero exponent property: any nonzero base to power 0 equals 1.

Flashcard 45: What is the value of $8^0$?

Answer:

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

Flashcard 46: Simplify: $\frac{\text{√}50}{\text{√}2}$

Answer: $5$. Use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ to get $\sqrt{25} = 5$.

Flashcard 47: What is the product rule for exponents with the same base, $a^m \cdot a^n$?

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

Flashcard 48: Simplify $\text{square root of } 100$.

Answer:

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

Flashcard 49: Rationalize the denominator: $\frac{4}{1+\sqrt{2}}$.

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

Flashcard 50: Find the value of $5^0$.

Answer:

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

Flashcard 51: What is the value of $8^0$?

Answer:

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

Flashcard 52: Identify the domain of $f(x)=\sqrt{\frac{x+1}{x-2}}$.

Answer: $(-\infty,-1]\cup(2,\infty)$. Need $\frac{x+1}{x-2} \ge 0$ with $x \ne 2$.

Flashcard 53: Identify the simplified value of $\sqrt{18x^2}$ for real $x$.

Answer: $3\sqrt{2}\,|x|$. $\sqrt{18x^2} = \sqrt{9 \cdot 2 \cdot x^2} = 3|x|\sqrt{2}$.

Flashcard 54: Find the value of $5^0$.

Answer:

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

Flashcard 55: What is the value of $2^{-3}$?

Answer: $\frac{1}{8}$. Negative exponents mean reciprocal: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$.

Flashcard 56: What is $2^3$?

Answer:

1. Multiply 2 by itself 3 times: $2 \times 2 \times 2$.

Flashcard 57: What is the definition of a negative exponent, $a^{-n}$ for $a \ne 0$?

Answer: $\frac{1}{a^n}$. Negative exponent means reciprocal with positive exponent.

Flashcard 58: Rationalize the denominator: $\frac{3}{2\sqrt{5}}$.

Answer: $\frac{3\sqrt{5}}{10}$. Multiply by $\frac{\sqrt{5}}{\sqrt{5}}$ to eliminate radical in denominator.

Flashcard 59: Express $64$ as a power of $4$.

Answer: $4^3$. Find what power of 4 gives 64: $4 \times 4 \times 4 = 64$.

Flashcard 60: What is the value of $10^0$?

Answer:

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

Flashcard 61: What is the $y$-intercept of $f(x)=a^x$ for $a>0$?

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

Flashcard 62: Simplify: $\text{cube root of } 125$.

Answer:

1. $\sqrt[3]{125} = 5$ since $5^3 = 125$.

Flashcard 63: What is $3^4$?

Answer:

1. $3$ raised to the $4$th power equals $81$.

Flashcard 64: Simplify $(\text{√}3)^2$.

Answer:

1. Squaring a square root returns the original number.

Flashcard 65: Simplify: $\text{square root of } 25$.

Answer:

1. $\sqrt{25} = 5$ since $5^2 = 25$.

Flashcard 66: What is $4^3$?

Answer:

1. $4$ cubed means $4 \times 4 \times 4 = 64$.

Flashcard 67: Evaluate $(-2)^3$.

Answer: -8. Cube the negative number: $(-2) \times (-2) \times (-2)$.

Flashcard 68: Simplify $\frac{\text{√}75}{\text{√}3}$.

Answer: $5$. Use $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ to get $\sqrt{25} = 5$.

Flashcard 69: State the formula for exponential growth.

Answer: $y = a(1 + r)^t$. Where $a$ is initial value, $r$ is growth rate, $t$ is time.

Flashcard 70: Simplify: $\text{square root of } 144$.

Answer:

1. $\sqrt{144} = 12$ since $12^2 = 144$.

Flashcard 71: Simplify $(9^{\frac{1}{2}})^2$.

Answer:

1. Power to a power rule: $(a^m)^n = a^{mn}$, so $\frac{1}{2} \times 2 = 1$.

Flashcard 72: Find $x$ if $x^2 = 36$.

Answer: 6 or -6. Taking the square root of both sides gives $x = \pm 6$.

Flashcard 73: What is the simplification of $\sqrt{ab}$ for $a \ge 0$ and $b \ge 0$?

Answer: $\sqrt{a}\sqrt{b}$. Square root of product equals product of square roots.

Flashcard 74: What is $0.5$ as a power of $2$?

Answer: $2^{-1}$. Negative exponent means reciprocal: $\frac{1}{2} = 2^{-1}$.

Flashcard 75: Find $x$ if $x^2 = 36$.

Answer: 6 or -6. Taking the square root of both sides gives $x = \pm 6$.

Flashcard 76: State the property of exponents for raising a power to a power.

Answer: $(a^m)^n = a^{m \times n}$. When raising a power to a power, multiply the exponents.

Flashcard 77: Simplify: $\text{square root of } 144$.

Answer:

1. $\sqrt{144} = 12$ since $12^2 = 144$.

Flashcard 78: Simplify the expression $x^{\frac{2}{3}} \times x^{\frac{1}{3}}$.

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

Flashcard 79: What is $1$ as a power of any nonzero number?

Answer: $a^0$. Zero exponent property: any nonzero base to power 0 equals 1.

Flashcard 80: Simplify $\text{square root of } 100$.

Answer:

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

Flashcard 81: Simplify: $\text{square root of } 81$.

Answer:

1. $\sqrt{81} = 9$ since $9^2 = 81$.

Flashcard 82: What is the product rule for exponents with the same base, $a^m \cdot a^n$?

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

Flashcard 83: Simplify: $\text{cube root of } 64$.

Answer:

1. $\sqrt[3]{64} = 4$ since $4^3 = 64$.

Flashcard 84: What is the inverse relationship between $y=a^x$ and logarithms for $a>0$, $a\ne 1$?

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

Flashcard 85: Convert $\sqrt{49}$ to an exponential form.

Answer: $49^{\frac{1}{2}}$. Square root as fractional exponent with denominator 2.

Flashcard 86: Simplify: $\text{cube root of } 125$.

Answer:

1. $\sqrt[3]{125} = 5$ since $5^3 = 125$.

Flashcard 87: Simplify: $\text{square root of } 81$.

Answer:

1. $\sqrt{81} = 9$ since $9^2 = 81$.

Flashcard 88: What is the simplified value of $\sqrt{\frac{49}{64}}$?

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

Flashcard 89: Identify the simplified value of $\frac{5^7}{5^2}$.

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

Flashcard 90: What is the simplification identity for all real $x$, $\sqrt{x^2}$?

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

Flashcard 91: Simplify: $\frac{\text{square root of } 64}{\text{square root of } 16}$.

Answer:

1. $\frac{\sqrt{64}}{\sqrt{16}} = \frac{8}{4} = 2$.

Flashcard 92: State the value of $2^5$.

Answer:

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

Flashcard 93: What is $\text{√}(100)$?

Answer:

1. The positive number that when squared equals 100.

Flashcard 94: Simplify $x^{\frac{1}{2}} \times x^{\frac{1}{2}}$.

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

Flashcard 95: State the exponential form of 100.

Answer: $10^2$. $100 = 10 \times 10 = 10^2$.

Flashcard 96: What is the square of 9?

Answer:

1. $9^2 = 9 \times 9 = 81$.

Flashcard 97: What is the definition of a zero exponent for $a \neq 0$, $a^0$?

Answer:

1. Any nonzero number raised to the power of zero equals $1$.

Flashcard 98: What is the definition of a negative exponent, $a^{-n}$ for $a e 0$?

Answer: $\frac{1}{a^n}$. Negative exponent means reciprocal with positive exponent.

Flashcard 99: What is the value of $\text{√}1$?

Answer:

1. The square root of 1 is always 1.

Flashcard 100: Simplify: $\text{cube root of } 64$.

Answer:

1. $\sqrt[3]{64} = 4$ since $4^3 = 64$.