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PSAT Math : Algebra

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Example Questions

Example Question #9 : How To Multiply Complex Numbers

What is the eighth power of 1+ i ?

Possible Answers:

8 - 8i

8 + 8i

The correct response is not given among the other choices.

256i

Correct answer:

Explanation:

First, square 1+ i using the square of a binomial pattern as follows:

$\left (1+ i \right )^{2}$

$= 1^{2}+ 2 \cdot 1 \cdot i + i^{2}$

= 1 +2i -1

= 2i

Raising this number to the fourth power yields the correct response:

$\left (1+ i \right )^{8}$

$= \left [\left (1+ i \right )^{2} \right ]^{4}$

$= (2i)^{4}$

$= (2 )^{4} \cdot i^{4}$

$=16 \cdot 1 = 16$

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Example Question #10 : How To Multiply Complex Numbers

What is the ninth power of -2i ?

Possible Answers:

18i

-18i

None of the other responses is correct.

512 i

-512 i

Correct answer:

-512 i

Explanation:

$\left ( -2i\right )^{9}$

$= \left ( -2 \right )^{9} \cdot i^{9}$

To raise a negative number to an odd power, take the absolute value of the base to that power and give its opposite:

$\left ( -2\right )^{9} = - \left ( 2^{9}\right ) = -512$

To raise to a power, divide the power by 4 and raise to the remainder. Since

$9 \div 4 = 2 \textup{ R }1$ ,

$i^{9} = i^{1}= i$

Therefore,

$\left ( -2i\right )^{9}= \left ( -2 \right )^{9} \cdot i^{9} = -512 i$

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Example Question #11 : Complex Numbers

Simplify:

$\small \left ( 3 - 2i \right )\left ( 2 + i\right )$

Possible Answers:

$\small 6 + 3i$

$\small 6 - i$

$\small 8 + i$

$\small 6 - 3i$

$\small 8 - i$

Correct answer:

$\small 8 - i$

Explanation:

Use the FOIL method that states to multiply the Firsts, Outter, Inner, Lasts. Also remember that $\small i \times i = -1$ :

$\small \left ( 3 - 2i \right )\left ( 2 + i\right )$

$\small = 3 \times 2 + 3 \times i - 2i \times 2 - 2i \times i$

$\small = 6 +3i -4i + 2$

$\small = 8 - i$

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Example Question #91 : Exponents

If and are positive integers and , then what is the value of $\frac{m}{n}$ ?

Possible Answers:

$\frac{1}{3}$

$\frac{1}{16}$

$\frac{5}{3}$

Correct answer:

Explanation:

4³= 64

Alternatively written, this is 4(4)(4) = 64 or 4³= 64^1.

Thus, m = 3 and n = 1.

m/n = 3/1 = 3.

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Example Question #531 : Algebra

Write the following logarithm in expanded form:

$\log x^{2}y$

Possible Answers:

$2\left ( \log xy \right )$

$2\log x+\log y$

$\log x^{2}+\log y$

$\log x+\log y$

$2\log x-\log y$

Correct answer:

$2\log x+\log y$

Explanation:

$\log x^{2}y=\log x^{2}+\log y=2\log x+\log y$

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Example Question #1 : How To Find An Exponent From A Rational Number

Com_exp_1

Which of the following lists the above quantities from least to greatest?

Possible Answers:

I, IV, III, II

II, III, I, IV

I, III, II, IV

I, IV, II, III

IV, III, II, I

Correct answer:

I, III, II, IV

Explanation:

Com_exp_2

Com_exp_3

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Example Question #1 : Exponents And Rational Numbers

Solve for .

$2^{x}= 64$

Possible Answers:

x=64

x=-6

$x=64^{2}$

x=5

x=6

Correct answer:

x=6

Explanation:

Since $2^{x}= 2^{6}$

Hence x=6

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Example Question #1 : Exponents And Rational Numbers

Simplify:

$\sqrt[3]{x^{10}y^{8}z^{9}}$

Possible Answers:

$x^{3}y^{3}z^{3}$

$x^{3}y^{2}z^{3}$

$x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$

$x^{3}y^{2}z^{3}^{\sqrt[3]{xyz}}$

$\sqrt[3]{xy^{2}}$

Correct answer:

$x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$

Explanation:

$\sqrt[3]{x^{9}xy^{6}y^{2}z^{9}}=x^{3}y^{2}z^{3}\sqrt[3]{xy^{2}}$

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Example Question #4 : How To Find An Exponent From A Rational Number

Solve for :

$\sqrt{x} -1= 4$

Possible Answers:

Correct answer:

Explanation:

From the equation one can see that

$\sqrt{x} = 5$

Hence must be equal to 25.

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Example Question #2 : Exponents And Rational Numbers

Evaluate:

$\log 0.1=$

Possible Answers:

$\frac{1}{10}$

Correct answer:

Explanation:

$0.1= 10^{-1}$

$\log 0.1=-1$

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PSAT Math : Algebra

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #9 : How To Multiply Complex Numbers

Example Question #10 : How To Multiply Complex Numbers

Example Question #11 : Complex Numbers

Example Question #91 : Exponents

Example Question #531 : Algebra

Example Question #1 : How To Find An Exponent From A Rational Number

Example Question #1 : Exponents And Rational Numbers

Example Question #1 : Exponents And Rational Numbers

Example Question #4 : How To Find An Exponent From A Rational Number

Example Question #2 : Exponents And Rational Numbers

All PSAT Math Resources

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