SAT MAth Practice questions – all topics
- Advanced Math Weightage: 35% Questions: 13-15
- Equivalent expressions
- Nonlinear equations in one variable and systems of equations in two variables
- Nonlinear functions
SAT MAth and English – full syllabus practice tests
Question Hard
19
$
\sqrt{14-2 x}=x-7
$
What value of \(\mathrm{x}\) satisfies the given equation?
▶️Answer/Explanation
Ans:7
To solve the equation \(\sqrt{14 – 2x} = x – 7\) for \(x\):
Square both sides to eliminate the square root:
\[
(\sqrt{14 – 2x})^2 = (x – 7)^2
\]
\[
14 – 2x = (x – 7)^2
\]
Expand the right-hand side:
\[
14 – 2x = x^2 – 14x + 49
\]
Rearrange the equation to set it to zero:
\[
x^2 – 14x + 49 + 2x – 14 = 0
\]
\[
x^2 – 12x + 35 = 0
\]
Factor the quadratic equation:
\[
(x – 5)(x – 7) = 0
\]
Solve for \(x\):
\[
x – 5 = 0 \quad \text{or} \quad x – 7 = 0
\]
\[
x = 5 \quad \text{or} \quad x = 7
\]
Check both solutions in the original equation:
For \(x = 5\):
\[
\sqrt{14 – 2(5)} = 5 – 7
\]
\[
\sqrt{4} = -2 \quad (\text{Not true, so } x = 5 \text{ is not a solution})
\]
For \(x = 7\):
\[
\sqrt{14 – 2(7)} = 7 – 7
\]
\[
\sqrt{0} = 0 \quad (\text{True, so } x = 7 \text{ is a solution})
\]
The value of \(x\) that satisfies the equation is \(7\).
Question Hard
Which of the following expressions is equivalent to \({(2\sqrt{x}-\sqrt{y})}^{\frac{2}{5}}\) , where x>y and y>0?
A) \({(4x-y)}^5\)
B) \(\sqrt[5]{4x-y}\)
C) \({(4x-4 \sqrt{xy}+y)}^{\frac{1}{5}}\)
D) \(\sqrt[5]{4x-4xy+y}\)
▶️Answer/Explanation
C) \({(4x-4\sqrt{xy}+y)}^{\frac{1}{5}}\)
We need to determine the expression equivalent to \((2 \sqrt{x} – \sqrt{y})^{\frac{2}{5}}\).
1. Simplify the expression inside the parentheses:
Given: \((2 \sqrt{x} – \sqrt{y})^{\frac{2}{5}}\)
2. Examine each option:
Option A: \((4 x – y)^5\)
\[
(4x – y)^{\frac{1}{5}} \neq (2\sqrt{x} – \sqrt{y})^{\frac{2}{5}}
\]
Option B: \(\sqrt[5]{4 x – y}\)
\[
(4x – y)^{\frac{1}{5}} \neq (2\sqrt{x} – \sqrt{y})^{\frac{2}{5}}
\]
Option C: \((4 x – 4 \sqrt{x y} + y)^{\frac{1}{5}}\)
\[
(4x – 4\sqrt{xy} + y)^{\frac{1}{5}} = (2\sqrt{x} – \sqrt{y})^{\frac{2}{5}}
\]
Option D: \(\sqrt[5]{4 x – 4xy + y}\)
\[
(4x – 4xy + y)^{\frac{1}{5}} \neq (2\sqrt{x} – \sqrt{y})^{\frac{2}{5}}
\]
Thus, the correct expression is:
\[ \boxed{(4 x – 4 \sqrt{x y} + y)^{\frac{1}{5}}} \]
Question Hard
Which of the following is equivalent to \((\sqrt{32})(\sqrt[5]{64})\) ?
A. \(6\left(\sqrt[7]{2^5}\right)\)
B. \(6\left(\sqrt[10]{2^7}\right)\)
c. \(8\left(\sqrt[7]{2^5}\right)\)
D. \(8\left(\sqrt[10]{2^7}\right)\)
▶️Answer/Explanation
Ans:D
To simplify \((\sqrt{32})(\sqrt[5]{64})\), we can first rewrite the numbers under the square roots as powers of 2 :
\[
\begin{aligned}
& \sqrt{32}=\sqrt{2^5}=2^{\frac{5}{2}} \\
& \sqrt[5]{64}=\sqrt[5]{2^6}=2^{\frac{6}{5}}
\end{aligned}
\]
Now, when we multiply these together, we add the exponents:
\[
(\sqrt{32})(\sqrt[5]{64})=2^{\frac{5}{2}} \cdot 2^{\frac{6}{5}}=2^{\frac{5}{2}+\frac{6}{5}}=2^{\frac{25}{10}+\frac{12}{10}}=2^{\frac{37}{10}}
\]
A. \(6\left(\sqrt[7]{2^5}\right)\)
\[6\left(\sqrt[7]{2^5}\right) = 6 \times 2^{\frac{5}{7}}\]
This expression cannot be simplified to \(2^{\frac{37}{10}}\).
B. \(6\left(\sqrt[10]{2^7}\right)\)
\[6\left(\sqrt[10]{2^7}\right) = 6 \times 2^{\frac{7}{10}}\]
This expression cannot be simplified to \(2^{\frac{37}{10}}\).
C. \(8\left(\sqrt[7]{2^5}\right)\)
\[8\left(\sqrt[7]{2^5}\right) = 8 \times 2^{\frac{5}{7}}\]
This expression cannot be simplified to \(2^{\frac{37}{10}}\).
D. \(8\left(\sqrt[10]{2^7}\right)\)
\[8\left(\sqrt[10]{2^7}\right) = 8 \times 2^{\frac{7}{10}}\]
This expression can be simplified to \(2^{\frac{37}{10}}\), since \(8 \times 2^{\frac{7}{10}} = 2^{\frac{3}{10}} \times 2^{\frac{7}{10}} = 2^{\frac{3}{10} + \frac{7}{10}} = 2^{\frac{37}{10}}\).
So, after solving each option, we see that only option D simplifies to \(2^{\frac{37}{10}}\).
Question Hard
Which of the following expressions is equivalent to $(\sqrt{2 q}+\sqrt{2 r})^{\frac{2}{3}}$, where $q>0$ and $r<0$ ?
A) $(2 q+2 r)^3$
B) $\sqrt[3]{2 q+2 r}$
C) $\sqrt[3]{2 q+2 \sqrt{q r}+2 r}$
D) $\sqrt[3]{2 q+4 \sqrt{q r}+2 r}$
▶️Answer/Explanation
D
Question
In 480 BC, the population of the Persian Empire was approximately 49.4 million. The population of the Persian Empire was 44% of the world population at that time. Which of the following is the best estimate of the world population in 480 BC?
- 21.7 million
- 89.1 million
- 93.4 million
- 112.3 million
▶️Answer/Explanation
D
Question Hard
1. Which expression is equivalent to \( (x^2 y)(x^4 y^{-3}) \), where \( x, y, \) and \( z \) are positive numbers?
A) \( x^6 y^{-3} \)
B) \( x^6 y^{-2} \)
C) \( x^4 y^{-3} \)
D) \( x^2 y^{-2} \)
▶️Answer/Explanation
Answer: B
Use exponent rules:
When multiplying terms with the same base, add the exponents.
Simplify \( x \)-terms:
$
x^2 \cdot x^4 = x^{2+4} = x^6
$
Simplify \( y \)-terms:
$
y \cdot y^{-3} = y^{1+(-3)} = y^{-2}
$
Final expression:
$
x^6 y^{-2}
$
Question Hard
2.Which of the following is equivalent to \( \sqrt{\frac{x}{64}} \) for all \( x > 0 \)?
A) \( \frac{x^2}{8} \)
B) \( \frac{x^2}{32} \)
C) \( \frac{x^{\frac{1}{2}}}{8} \)
D) \( \frac{x^{\frac{1}{2}}}{32} \)
▶️Answer/Explanation
Answer: C
Simplify \( \sqrt{\frac{x}{64}} \):
$
\sqrt{\frac{x}{64}} = \frac{\sqrt{x}}{\sqrt{64}} = \frac{x^{\frac{1}{2}}}{8}
$
Question Hard
3. Which expression is equivalent to \( \frac{x^{\frac{5}{2}}}{\sqrt{x}} \), where \( x \neq 0 \)?
A) \( x^{\frac{3}{2}} \)
B) \( x^{\frac{5}{3}} \)
C) \( x^{\frac{3}{2}} \)
D) \( x^2 \)
▶️Answer/Explanation
Answer: D
Simplify:
$
\frac{x^{\frac{5}{2}}}{\sqrt{x}} = \frac{x^{\frac{5}{2}}}{x^{\frac{1}{2}}}
$
Apply exponent rule (\( a^m / a^n = a^{m-n} \)):
$
x^{\frac{5}{2} – \frac{1}{2}} = x^{\frac{4}{2}} = x^2
$
Question Hard
4. Which expression is equivalent to \( b^{\frac{y}{x}} \), where \( b > 0 \)?
A) \( \sqrt[5]{b^{105}} \)
B) \( \sqrt[5]{b^{135}} \)
C) \( \sqrt[105]{b^{135}} \)
D) \( \sqrt[135]{b^{105}} \)
▶️Answer/Explanation
Answer: C
Convert to Radical Form:
$
b^{\frac{y}{x}} = \sqrt[x]{b^y}
$
The correct form is \( \sqrt[105]{b^{135}} \), since \( \frac{135}{105} = \frac{y}{x} \).
Question Hard
5. Which expression represents the product of \( (b^6 c^{-2} d^{-5}) \) and \( (b^8 c^{-3} + c^4 d^5) \), where \( b, c, \) and \( d \) are positive?
A) \( b^{14} c^{-5} + c^2 d^{-10} \)
B) \( b^{14} c^{-5} + c^2 \)
C) \( b^{14} c^{-5} d^{-5} + b^6 c^2 \)
D) \( b^{14} c^{-5} d^{-5} + c^2 \)
▶️Answer/Explanation
Answer: C
Multiply the Terms:
$
(b^6 c^{-2} d^{-5}) \cdot (b^8 c^{-3} + c^4 d^5)
$
Distribute:
$
b^6 c^{-2} d^{-5} \cdot b^8 c^{-3} + b^6 c^{-2} d^{-5} \cdot c^4 d^5
$
Simplify Each Term:
1. First term:
$
b^{6+8} c^{-2-3} d^{-5} = b^{14} c^{-5} d^{-5}
$
2. Second term:
$
b^6 c^{-2+4} d^{-5+5} = b^6 c^2 d^0 = b^6 c^2
$
Question Hard
6. If \( x \neq 0 \) and \( y \neq 0 \), which of the following is equivalent to \( \frac{8x^2}{\sqrt{4x^6 y^4}} \)?
A) \( 2xy^{-2} \)
B) \( 4x^{-2} y^2 \)
C) \( 4x^{-1} y^{-2} \)
D) \( 4xy^2 \)
▶️Answer/Explanation
Answer:
Question Hard
7. If \( r \) and \( s \) are positive, which of the following expressions is equivalent to \( \frac{r^3}{s^7} \)?
A) \( \frac{1}{\sqrt[7]{r^5 s^{14}}} \)
B) \( \sqrt[7]{r^5 s^{14}} \)
C) \( \frac{1}{\sqrt[7]{r^6 s^3}} \)
D) \( \sqrt[7]{r^9 s^3} \)
▶️Answer/Explanation
Answer:
Question Hard
8. The given equation \( \sqrt[d]{a} = \sqrt[b]{c} \) relates the distinct positive numbers \( a, b, c, \) and \( d \). Which equation correctly expresses \( c \) in terms of \( a, b, \) and \( d \)?
A) \( c = \frac{d}{a^{\frac{1}{b}}} \)
B) \( c = a^{d – h} \)
C) \( c = a^{\frac{d}{b}} \)
D) \( c = a^{\frac{b}{d}} \)
▶️Answer/Explanation
Answer:
Question Hard
9. Which expression is equivalent to \( \sqrt{16a^3} \), where \( a > 0 \)?
A) \( 4a^{\frac{3}{2}} \)
B) \( 4a^{\frac{5}{3}} \)
C) \( 8a^{\frac{3}{4}} \)
D) \( 8a^{\frac{4}{3}} \)
▶️Answer/Explanation
Answer:
Question Hard
10. Which expression is equivalent to \( \sqrt{r s} (\sqrt{r} + \sqrt{s}) \), where \( r \geq 0 \) and \( s \geq 0 \)?
A) \( \sqrt{r^2 s} + r s^{\frac{5}{2}} \)
B) \( r \sqrt{s} + s \sqrt{r} \)
C) \( rs \sqrt{r} + s \)
D) \( \sqrt{rs} + r + s \)
▶️Answer/Explanation
Answer:
Question Hard
11. The expression \( \left( \sqrt{x^2} \right)^n \), where \( n \) is a constant, is equivalent to \( x^8 \). What is the value of \( n \)?
▶️Answer/Explanation
Answer:
Question Hard
12. Which expression is equivalent to \( 2^{-2k} \cdot 3^k \)?
A) \( (3\sqrt{2})^k \)
B) \( \left( \frac{1}{36} \right)^k \)
C) \( \left( \frac{3}{4} \right)^k \)
D) \( \left( \frac{9}{4} \right)^k \)
▶️Answer/Explanation
Answer:
Question Hard
13. Which expression is equivalent to \( a^{\frac{2}{6}} \cdot (a^{\frac{5}{3}})^{\frac{1}{2}} \), where \( a \) is positive?
A) \( \sqrt[3]{a^2} \)
B) \( \sqrt[3]{a^{16}} \)
C) \( \sqrt[3]{a^8} \)
D) \( \sqrt[3]{a^{14}} \)
▶️Answer/Explanation
Answer: