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SAT Math : Algebra

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Example Questions

Example Question #831 : Algebra

Simplify:

$\frac{\frac{a^2b^3c}{a^5b^7c^4}}{abc}$

Possible Answers:

$\frac{1}{a^4b^5c^4}$

$\frac{a}{b}$

$\frac{a^5b^7c^4}{ab^2}$

None of the given answers

a^2b^3c

Correct answer:

$\frac{1}{a^4b^5c^4}$

Explanation:

We can rewrite this problem so that it's a simpler multiplication problem. Take the bottom of the express ( abc ) and write its reciprocal, then multiply it by the fraction in the numerator, like so:

$\frac{\frac{a^2b^3c}{a^5b^7c^4}}{abc} = \frac{a^2b^3c}{a^5b^7c^4} \cdot \frac{1}{abc}$

Now, we can simplify using cross-division.

$\frac{a^2b^3c}{a^5b^7c^4} \cdot \frac{1}{abc} = \frac{ab^2}{a^5b^7c^4}$

Now, we can simplify this expression. When we divide these terms, we subtract their exponents.

$\frac{ab^2}{a^5b^7c^4} = \frac{1}{a^4b^5c^4}$

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Example Question #69 : How To Simplify An Expression

Simplify.

$\frac{a^{4}b^{-2}c^{5}d^{14}e^{10}}{a^{6}b^{3}c^{7}d^{-7}e^{8}}$

Possible Answers:

$\frac{d^{7}e^{2}}{a^{2}b^{1}c^{2}}$

$a^{10}b^{1}c^{12}d^{7}e^{18}$

$\frac{d^{21}e^{2}}{a^{2}b^{5}c^{2}}$

$a^{\frac{2}3{}}b^{\frac{-2}{3}}c^{\frac{5}{7}}d^{-2}e^{\frac{5}{4}}$

Correct answer:

$\frac{d^{21}e^{2}}{a^{2}b^{5}c^{2}}$

Explanation:

Separate each group of variables to simplify.

$\frac{a^{4}}{a^{6}} =$ $\frac{a\times a\times a\times a}{a \times a \times a \times a \times a \times a}$ $= \frac{1}{a^{2}}$

$\frac{b^{-2}}{b^{3}}=$ $\frac{1}{b^{2}}\times\frac{1}{b^{3}}=$ $\frac{1}{b\times b\times b\times b\times b}=\frac{1}{b^{5}}$

$\frac{c^{5}}{c^{7}} =$ $\frac{c\times c\times c\times c\times c}{c\times c\times c\times c\times c\times c\times c}$ $= \frac{1}{c^{2}}$

$\frac{d^{14}}{d^{-7}}=\frac{d^{14}}{1}\times \frac{d^{7}}{1}$ $= \frac{d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d\times d}{1}$

$= \frac{d^{21}}{1}$

$\frac{e^{10}}{e^{8}}=\frac{e\times e\times e\times e\times e\times e\times e\times e\times e\times e}{e\times e\times e\times e\times e\times e\times e\times e}$ $= \frac{e^{2}}{1}$

$\frac{1}{a^{2}}\times \frac{1}{b^{5}}\times \frac{1}{c^{2}}\times \frac{d^{21}}{1}\times \frac{e^{2}}{1}= \frac{d^{21}e^{2}}{a^{2}b^{5}c^{2}}$

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Example Question #71 : How To Simplify An Expression

$\frac{1}{u+1} = t$ . $u \ne -1, u \ne 4$ .

Which of the following is equal to $\frac{1}{u-4}$ ?

Possible Answers:

t+5

$\frac{1-5t}{t}$

$\frac{t}{t-5}$

$\frac{t}{1-5t}$

$\frac{5t}{1-t}$

Correct answer:

$\frac{t}{1-5t}$

Explanation:

Take the reciprocal of both expressions:

$\frac{1}{u+1} = t$

$u+1 = \frac{1}{t}$

Subtract 5 from both sides:

$u+1-5 = \frac{1}{t}-5$

$u-4 = \frac{1}{t}-5$

Rewrite the expression at left and simplify it:

$u-4 = \frac{1}{t}-\frac{5t}{t}$

$u-4 = \frac{1-5t}{t}$

Take the reciprocal of both expressions:

$\frac{1}{u-4} = \frac{t}{1-5t}$

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Example Question #2602 : Sat Mathematics

$\sqrt{x+ 1} = t$ ; $x \ge 7$

Which of the following is equal to $\sqrt{x- 7}$ ?

Possible Answers:

$\sqrt{t^{2} -8}$

$t -1 - \sqrt{7}$

$t - 2 \sqrt{2}$

$\sqrt{t^{2} -64 }$

t -8

Correct answer:

$\sqrt{t^{2} -8}$

Explanation:

Square both expressions

$(\sqrt{x+ 1} )^{2}= t^{2}$

$x+1 = t^{2}$

Subtract 8 from both sides:

$x+1-8 = t^{2} -8$

$x-7= t^{2} - 8$

Take the positive square root of both sides:

$\sqrt{x-7}= \sqrt{t^{2} -8}$

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Example Question #143 : Expressions

$\sqrt{x+ 1} = t$ for $x \ge -1$ .

Which of the following is equal to $\sqrt{x+ 7}$ ?

Possible Answers:

$t-1+ \sqrt{7}$

t+ 6

$t+ \sqrt{6}$

$\sqrt{t^{2} + 3 6}$

$\sqrt{t^{2} + 6}$

Correct answer:

$\sqrt{t^{2} + 6}$

Explanation:

Square both expressions

$(\sqrt{x+ 1} )^{2}= t^{2}$

$x+1 = t^{2}$

Add 6 to both sides:

$x+1+ 6 = t^{2} + 6$

$x+7= t^{2} + 6$

Take the square root of both sides:

$\sqrt{x+7}= \sqrt{t^{2} + 6}$

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Example Question #144 : Expressions

$(x+1) ^{2} = t$ . is a positive number.

Which of the following is equal to $(x+ 5)^{2}$ ?

Possible Answers:

$t +12\sqrt{ t} +36$

$t -12\sqrt{ t} +36$

$t + 8\sqrt{ t} +16$

None of the other choices gives the correct response.

$t -8\sqrt{ t} +16$

Correct answer:

$t + 8\sqrt{ t} +16$

Explanation:

$(x+1) ^{2} = t$ , so, taking the square root of both sides:

$x+1 = \pm\sqrt{ t}$

is positive, so x+1 is as well; therefore,

$x+1 = \sqrt{ t}$

Add 4 to both sides:

$x+1 + 4 = \sqrt{ t} + 4$

$x+5 =\sqrt{ t} + 4$

Square both sides, and apply the binomial square pattern to the right expression:

$(x+5 )^{2}= (\sqrt{ t} + 4)^{2}$

$= (\sqrt{ t})^{2}+ 2\cdot4 \cdot (\sqrt{ t}) +4 ^{2}$

$=t + 8\sqrt{ t} +16$

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Example Question #145 : Expressions

$\frac{1}{u+1} = t$ . $u \ne -1, u \ne -5$ .

Which of the following is equal to $\frac{1}{u+5}$ ?

Possible Answers:

$\frac{4}{1+4t}$

$\frac{t}{1+4t}$

$\frac{t+4}{4}$

$\frac{t}{t+4 }$

$\frac{4}{t+4 }$

Correct answer:

$\frac{t}{1+4t}$

Explanation:

Take the reciprocal of both expressions:

$\frac{1}{u+1} = t$

$u+1 = \frac{1}{t}$

Add 4 to both sides:

$u+1+ 4 = \frac{1}{t}+ 4$

$u+5 = \frac{1}{t}+ 4$

Rewrite the expression at left and simplify it:

$u+5 = \frac{1}{t}+ \frac{4t}{t}$

$u+5 = \frac{1+4t}{t}$

Take the reciprocal of both expressions:

$\frac{1}{u+5} = \frac{t}{1+4t}$

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Example Question #23 : How To Simplify An Expression

Simplify the following expression: x³ - 4(x² + 3) + 15

Possible Answers:

x^3-4x^2+3

x^3-12x^2+15

x^3-3x^2+15

x^5+3

x^3-4x^2+3

Correct answer:

x^3-4x^2+3

Explanation:

To simplify this expression, you must combine like terms. You should first use the distributive property and multiply -4 by x²and -4 by 3.

x³ - 4x² -12 + 15

You can then add -12 and 15, which equals 3.

You now have x³ - 4x² + 3 and are finished. Just a reminder that x³ and 4x²are not like terms as the x’s have different exponents.

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Example Question #1 : How To Simplify A Fraction

The expression $(\frac{a^{2}}{b^{3}})(\frac{a^{-2}}{b^{-3}}) = ?$

Possible Answers:

$0$

$b^{-9}$

$1$

$\frac{a^{-4}}{b^{-9}}$

$\frac{b^{9}}{a^{4}}$

Correct answer:

$1$

Explanation:

A negative exponent in the numerator of a fraction can be rewritten with a positive exponent in the denominator. The same is true for a negative exponent in the denominator. Thus, $\frac{a^{-2}}{b^{-3}} =\frac{b^{3}}{a^{2}}$ .

When $\frac{a^{2}}{b^{3}}$ is multiplied by $\frac{b^{3}}{a^{2}}$ , the numerators and denominators cancel out, and you are left with 1.

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Example Question #2 : How To Simplify A Fraction

Two two-digit numbers, 'AB' and 'BA' , sum to produce a three-digit number in which the second digit is equal to . The addition is represented below. (Note that the variables are used to represent individual digits; no multiplication is taking place).

AB + BA = 1A2

What is ?

Possible Answers:

Correct answer:

Explanation:

Another way to represent this question is:

$\begin{matrix} & \ \ AB\\ +& \ \ BA\\ & 1A2 \end{matrix}$

In the one's column, and add to produce a number with a two in the one's place. In the ten's column, we can see that a one must carry in order to get a digit in the hundred's place. Together, we can combine these deductions to see that the sum of and must be twelve (a one in the ten's place and a two in the one's place).

In the one's column: B+A=12

The one carries to the ten's column.

In the ten's column: (1)+A+B=(1)+12=13

The three goes into the answer and the one carries to the hundred's place. The final answer is 132. From this, we can see that {A=3} because 1A2=132 .

Using this information, we can solve for .

$A+B=12\ \text{and}\ A=3$

3+B=12

B=9

You can check your answer by returning to the original addition and plugging in the values of and .

$AB=39\ \text{and}\ BA=93$

AB+BA=39+93=132

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SAT Math : Algebra

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #831 : Algebra

Example Question #69 : How To Simplify An Expression

Example Question #71 : How To Simplify An Expression

Example Question #2602 : Sat Mathematics

Example Question #143 : Expressions

Example Question #144 : Expressions

Example Question #145 : Expressions

Example Question #23 : How To Simplify An Expression

Example Question #1 : How To Simplify A Fraction

Example Question #2 : How To Simplify A Fraction

All SAT Math Resources

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