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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 $\displaystyle 1+ i$ ?

Possible Answers:

8 - 8i $\displaystyle 8 - 8i$

$\displaystyle 16$

8 + 8i $\displaystyle 8 + 8i$

The correct response is not given among the other choices.

256i $\displaystyle 256i$

Correct answer:

$\displaystyle 16$

Explanation:

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

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

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

$\displaystyle = 1 +2i -1$

= 2i $\displaystyle = 2i$

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

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

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

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

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

$=16 \cdot 1 = 16$ $\displaystyle =16 \cdot 1 = 16$

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Example Question #10 : How To Multiply Complex Numbers

What is the ninth power of -2i $\displaystyle -2i$ ?

Possible Answers:

18i $\displaystyle 18i$

-18i $\displaystyle -18i$

None of the other responses is correct.

512 i $\displaystyle 512 i$

-512 i $\displaystyle -512 i$

Correct answer:

-512 i $\displaystyle -512 i$

Explanation:

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

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

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

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

To raise $\displaystyle i$ to a power, divide the power by 4 and raise $\displaystyle i$ to the remainder. Since

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

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

Therefore,

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

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Example Question #11 : Complex Numbers

Simplify:

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

Possible Answers:

$\small 6 + 3i$ $\displaystyle \small 6 + 3i$

$\small 6 - i$ $\displaystyle \small 6 - i$

$\small 8 + i$ $\displaystyle \small 8 + i$

$\small 6 - 3i$ $\displaystyle \small 6 - 3i$

$\small 8 - i$ $\displaystyle \small 8 - i$

Correct answer:

$\small 8 - i$ $\displaystyle \small 8 - i$

Explanation:

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

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

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

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

$\small = 8 - i$ $\displaystyle \small = 8 - i$

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Example Question #812 : Algebra

If $\displaystyle m$ and $\displaystyle n$ are positive integers and $\displaystyle 4^m=64^n$, then what is the value of $\frac{m}{n}$ $\displaystyle \frac{m}{n}$?

Possible Answers:

$\frac{5}{3}$ $\displaystyle \frac{5}{3}$

$\frac{1}{16}$ $\displaystyle \frac{1}{16}$

$\displaystyle 16$

$\frac{1}{3}$ $\displaystyle \frac{1}{3}$

$\displaystyle 3$

Correct answer:

$\displaystyle 3$

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:

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

Possible Answers:

$\log x+\log y$ $\displaystyle \log x+\log y$

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

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

$2\log x+\log y$ $\displaystyle 2\log x+\log y$

$2\log x-\log y$ $\displaystyle 2\log x-\log y$

Correct answer:

$2\log x+\log y$ $\displaystyle 2\log x+\log y$

Explanation:

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

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Example Question #7 : Exponents And Rational Numbers

Com_exp_1

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

Possible Answers:

I, IV, III, II

IV, III, II, I

I, III, II, IV

I, IV, II, III

II, III, I, IV

Correct answer:

I, III, II, IV

Explanation:

Com_exp_2

Com_exp_3

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Example Question #2 : Exponential Ratios And Rational Numbers

Solve for $\displaystyle x$.

$2^{x}= 64$ $\displaystyle 2^{x}= 64$

Possible Answers:

x=-6 $\displaystyle x=-6$

x=5 $\displaystyle x=5$

$x=64^{2}$ $\displaystyle x=64^{2}$

x=64 $\displaystyle x=64$

x=6 $\displaystyle x=6$

Correct answer:

x=6 $\displaystyle x=6$

Explanation:

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

Hence x=6 $\displaystyle x=6$

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Example Question #1 : Exponents And Rational Numbers

Simplify:

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

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Example Question #221 : Exponents

Solve for $\displaystyle x$:

$\sqrt{x} -1= 4$ $\displaystyle \sqrt{x} -1= 4$

Possible Answers:

$\displaystyle 9$

$\displaystyle 5$

$\displaystyle 25$

$\displaystyle 36$

Correct answer:

$\displaystyle 25$

Explanation:

From the equation one can see that

$\sqrt{x} = 5$ $\displaystyle \sqrt{x} = 5$

Hence $\displaystyle x$ must be equal to 25.

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Example Question #2 : Exponents And Rational Numbers

Evaluate:

$\log 0.1=$ $\displaystyle \log 0.1=$

Possible Answers:

$\displaystyle 1$

$\displaystyle -2$

$\frac{1}{10}$ $\displaystyle \frac{1}{10}$

$\displaystyle 10$

$\displaystyle -1$

Correct answer:

$\displaystyle -1$

Explanation:

$0.1= 10^{-1}$ $\displaystyle 0.1= 10^{-1}$

$\log 0.1=-1$ $\displaystyle \log 0.1=-1$

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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 #812 : Algebra

Example Question #531 : Algebra

Example Question #7 : Exponents And Rational Numbers

Example Question #2 : Exponential Ratios And Rational Numbers

Example Question #1 : Exponents And Rational Numbers

Example Question #221 : Exponents

Example Question #2 : Exponents And Rational Numbers

All PSAT Math Resources

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