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PSAT Math : PSAT Mathematics

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Example Questions

Example Question #1151 : Psat Mathematics

Simplify: $2a^{2}b(ab^{2} - a^{2}b)$ $\displaystyle 2a^{2}b(ab^{2} - a^{2}b)$

Possible Answers:

$2a^{3}b^{3} + a^{2}b^{2}$ $\displaystyle 2a^{3}b^{3} + a^{2}b^{2}$

$2a^{2}b + 2a^{4}b^{2}$ $\displaystyle 2a^{2}b + 2a^{4}b^{2}$

$2a^{3}b^{3} - 2a^{4}b^{2}$ $\displaystyle 2a^{3}b^{3} - 2a^{4}b^{2}$

$2ab^{2} - 2a^{2}b$ $\displaystyle 2ab^{2} - 2a^{2}b$

$ab^{2} - a^{2}b$ $\displaystyle ab^{2} - a^{2}b$

Correct answer:

$2a^{3}b^{3} - 2a^{4}b^{2}$ $\displaystyle 2a^{3}b^{3} - 2a^{4}b^{2}$

Explanation:

Use the distributive property: $a(b+c)=ab+ac$ $\displaystyle a(b+c)=ab+ac$. When we multiply variables with exponents, we keep the same base and add the exponents: $a^{m}a^{n} = a^{m+n}$ $\displaystyle a^{m}a^{n} = a^{m+n}$

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Example Question #18 : Exponents

Simplify:

$(6x^{2})^{2}\times y^{2} + xyz =$ $\displaystyle (6x^{2})^{2}\times y^{2} + xyz =$

Possible Answers:

$36x^{4}y^{2}+xyz$ $\displaystyle 36x^{4}y^{2}+xyz$

$6x^{4}y^{2}z + xyz$ $\displaystyle 6x^{4}y^{2}z + xyz$

$36x^{5}y^{3}z$ $\displaystyle 36x^{5}y^{3}z$

$\textbf{cannot be simplified}$ $\displaystyle \textbf{cannot be simplified}$

$36x^{4}y^{2}$ $\displaystyle 36x^{4}y^{2}$

Correct answer:

$36x^{4}y^{2}+xyz$ $\displaystyle 36x^{4}y^{2}+xyz$

Explanation:

$(6x^{2})^{2} = 6^{2}x^{4} = 36x^{4}$ $\displaystyle (6x^{2})^{2} = 6^{2}x^{4} = 36x^{4}$

$36x^{4}\times y^{2} = 36x^{4}y^{2}$ $\displaystyle 36x^{4}\times y^{2} = 36x^{4}y^{2}$

We cannot combine $36x^{4}y^{2}$ $\displaystyle 36x^{4}y^{2}$ with $xyz$ $\displaystyle xyz$, so $(6x^{2})^{2}\times y^{2} + xyz =36x^{4}y^{2}+xyz$ $\displaystyle (6x^{2})^{2}\times y^{2} + xyz =36x^{4}y^{2}+xyz$.

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Example Question #19 : Exponents

If c^2=15 $\displaystyle c^2=15$, then what is the value of c^4 $\displaystyle c^4$?

Possible Answers:

225 $\displaystyle 225$

125 $\displaystyle 125$

$\displaystyle 75$

110 $\displaystyle 110$

$\sqrt{15}$ $\displaystyle \sqrt{15}$

Correct answer:

225 $\displaystyle 225$

Explanation:

c⁴ is equal to (c²)(c²).

We know c² = 15. Plugging in this value gives us c⁴ = (15)(15) = 225.

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Example Question #20 : Exponents

If $\displaystyle a$ and $\displaystyle b$ are nonzero numbers such that $\displaystyle a^2 = b^3$, which of the following is equivalent to a^6 $\displaystyle a^6$?

Possible Answers:

b^7 $\displaystyle b^7$

b^8 $\displaystyle b^8$

b^6 $\displaystyle b^6$

$b^{11}$ $\displaystyle b^{11}$

b^9 $\displaystyle b^9$

Correct answer:

b^9 $\displaystyle b^9$

Explanation:

For this problem, we need to make use of the property of exponents, which states that (x^y)^z = x^y^z.

We are given a² but are asked to find a⁶.

Let's raise both sides of the equation to the third power, so that we will end up with a⁶ on the left side.

(a²)³ = (b³)³

Now, according to the property of exponents mentioned before, we can multiply the exponents.

a^(2*3) = b^(3*3)

a⁶ = b⁹

The answer is b⁹.

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Example Question #21 : Exponential Operations

If $\displaystyle a$ and $\displaystyle b$ are positive integers and $2^{2a}(2^{2b})=16$ $\displaystyle 2^{2a}(2^{2b})=16$, what is the value of a +b $\displaystyle a +b$?

Possible Answers:

$\displaystyle 0$

$\displaystyle 3$

$\displaystyle -1$

$\displaystyle 1$

$\displaystyle 2$

Correct answer:

$\displaystyle 2$

Explanation:

The question tells us that 2^2a ( 2^2b)= 16.

We can rewrite 16 as 2⁴, giving us 2^2a ( 2^2b)= 2⁴.

When terms with the same base are multipled, their exponents can be added:

2^{(2a +2b)}= 2⁴

Since the base is the same on both sides of the equation, we can equate the exponents:

2a +2b = 4

2(a + b) = 4

a + b = 2

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Example Question #501 : Algebra

Simplify:

$\left ( \frac{x}{2t^{2}} \right )^{3}$ $\displaystyle \left ( \frac{x}{2t^{2}} \right )^{3}$

Possible Answers:

$\frac{x^{3}}{ 8 t^{6} }$ $\displaystyle \frac{x^{3}}{ 8 t^{6} }$

$\frac{ 8 t^{6} }{x^{3}}$ $\displaystyle \frac{ 8 t^{6} }{x^{3}}$

$\frac{x^{3}}{ 6 t^{6} }$ $\displaystyle \frac{x^{3}}{ 6 t^{6} }$

$\frac{ 6 t^{6} }{x^{3}}$ $\displaystyle \frac{ 6 t^{6} }{x^{3}}$

$\frac{x^{3}}{ 9t^{6} }$ $\displaystyle \frac{x^{3}}{ 9t^{6} }$

Correct answer:

$\frac{x^{3}}{ 8 t^{6} }$ $\displaystyle \frac{x^{3}}{ 8 t^{6} }$

Explanation:

Apply the the various properties of exponents:

$\left ( \frac{x}{2t^{2}} \right )^{3}$ $\displaystyle \left ( \frac{x}{2t^{2}} \right )^{3}$

$=\frac{x^{3}}{ \left ( 2t^{2} \right ) ^{3}}$ $\displaystyle =\frac{x^{3}}{ \left ( 2t^{2} \right ) ^{3}}$

$=\frac{x^{3}}{ 2^{3} \left ( t^{2} \right )^{3} }$ $\displaystyle =\frac{x^{3}}{ 2^{3} \left ( t^{2} \right )^{3} }$

$=\frac{x^{3}}{ 8 t^{2 \cdot 3} }$ $\displaystyle =\frac{x^{3}}{ 8 t^{2 \cdot 3} }$

$=\frac{x^{3}}{ 8 t^{6} }$ $\displaystyle =\frac{x^{3}}{ 8 t^{6} }$

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Example Question #1155 : Psat Mathematics

$\textup{If }4^{3}\times8^{2}=2^{y}\textup{, what is the value of }y\textup{ ?}$ $\displaystyle \textup{If }4^{3}\times8^{2}=2^{y}\textup{, what is the value of }y\textup{ ?}$

Possible Answers:

$\displaystyle 10$

$\displaystyle 12$

$\displaystyle 16$

$\displaystyle 28$

$\displaystyle 6$

Correct answer:

$\displaystyle 12$

Explanation:

$\textup{4 and 8 are powers of 2. Find the common base and set exponents equal:}$ $\displaystyle \textup{4 and 8 are powers of 2. Find the common base and set exponents equal:}$

$\left ( 2^{2}\right )^{3}\times \left ( 2^{3}\right )^{2}=2^{y}\: \; \; \; \; \; \;2^{6}\times 2^{6}=2^{y}$ $\displaystyle \left ( 2^{2}\right )^{3}\times \left ( 2^{3}\right )^{2}=2^{y}\: \; \; \; \; \; \;2^{6}\times 2^{6}=2^{y}$

$2^{12}=2^{y}\: \: \: \: \: \: \:12=y$ $\displaystyle 2^{12}=2^{y}\: \: \: \: \: \: \:12=y$

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Example Question #61 : Exponents

Simplify: $x^{5}\cdot x^{2}$ $\displaystyle x^{5}\cdot x^{2}$

Possible Answers:

$2x^{7}$ $\displaystyle 2x^{7}$

$x^{7}$ $\displaystyle x^{7}$

This expression cannot be simplified any further

$x^{10}$ $\displaystyle x^{10}$

$2x^{10}$ $\displaystyle 2x^{10}$

Correct answer:

$x^{7}$ $\displaystyle x^{7}$

Explanation:

When you are multiplying and the bases are the same, you add the exponents together. Because both bases are $\displaystyle x$ you add 5+2=7 $\displaystyle 5+2=7$ as your new exponent. You then keep the same base, $\displaystyle x$, to the 7th power.

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Example Question #63 : How To Multiply Exponents

Solve for $\displaystyle x$:

$\sqrt[3]{x-3} = 2$ $\displaystyle \sqrt[3]{x-3} = 2$

Possible Answers:

$None\ of\ the\ above$ $\displaystyle None\ of\ the\ above$

$\displaystyle 5$

$\displaystyle 22$

$\displaystyle 11$

$\displaystyle 7$

Correct answer:

$\displaystyle 11$

Explanation:

$\left ( \sqrt[3]{x-3} \right )^{3} = 2^{3}$ $\displaystyle \left ( \sqrt[3]{x-3} \right )^{3} = 2^{3}$

Now the left side equals x-3 $\displaystyle x-3$ and the right side equals 8. Hence:

x-3 = 8 $\displaystyle x-3 = 8$

Therefore $\displaystyle x$ must be equal to 11.

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Example Question #1152 : Psat Mathematics

Simplify the radical.

$\sqrt{3283}$ $\displaystyle \sqrt{3283}$

Possible Answers:

$7\sqrt{67}$ $\displaystyle 7\sqrt{67}$

$67\sqrt{49}$ $\displaystyle 67\sqrt{49}$

$57.3$ $\displaystyle 57.3$

$56$ $\displaystyle 56$

$7\sqrt{63}$ $\displaystyle 7\sqrt{63}$

Correct answer:

$7\sqrt{67}$ $\displaystyle 7\sqrt{67}$

Explanation:

We can break the square root down into 2 roots of 67 and 49. 49 is a perfect square and reduces to 7.

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PSAT Math : PSAT Mathematics

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1151 : Psat Mathematics

Example Question #18 : Exponents

Example Question #19 : Exponents

Example Question #20 : Exponents

Example Question #21 : Exponential Operations

Example Question #501 : Algebra

Example Question #1155 : Psat Mathematics

Example Question #61 : Exponents

Example Question #63 : How To Multiply Exponents

Example Question #1152 : Psat Mathematics

All PSAT Math Resources

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