How to multiply variables

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ISEE Middle Level Quantitative Reasoning › How to multiply variables

Questions 1 - 10

1

Simplify:

$(\frac{1}{4}x-\frac{3}{4})^{2}$

$\frac{x^2}{16}-\frac{3}{8}x+\frac{9}{16}$

$\frac{x^2}{16}+\frac{3}{8}x+\frac{9}{16}$

$\frac{x^2}{8}-\frac{3}{8}x+\frac{9}{16}$

$\frac{x^2}{16}-\frac{3}{16}x+\frac{9}{16}$

None of these

Explanation

We can solve it using the pattern for the square of a binominal or by using FOIL:

$(\frac{1}{4}x-\frac{3}{4})^{2}=(\frac{1}{4}x)^2-(2)( \frac{1}{4}x)( \frac{3}{4})+(\frac{3}{4})^2$

$=\frac{x^2}{16}-\frac{3}{8}x+\frac{9}{16}$

2

Simplify:

$(x^4)^{\frac{5}{2}}$

$x^{10}$

$x^{12}$

$x^{8}$

$x^{20}$

Explanation

We can apply the power of a power property:

$(x^4)^{\frac{5}{2}}=x^{(4\times \frac{5}{2})}=x^{10}$

3

$t^{2}*t^{2}=$

$t^{4}$

$t^{22}$

Cannot be simplified further

Explanation

Because the two terms have the same base, add the exponents and leave the base unchanged:

$t^{2}*t^{2}=t^{4}$

Answer: $t^{4}$

4

2 rabbits give birth to a litter of 8 rabbits. Each pair in that litter creates 4 rabbits. How many rabbits are there total, including the original 2 rabbits and their 8 children?

Explanation

If there are 8 rabbits, then that means there are 4 pairs of rabbits.

If each pair gives birth to 4 rabbits, that means there will be 16 rabbits produced by the litter ( $4\cdot4=16$ ).

Thus, the total number of rabbits is equal to .

Thus, 26 is the correct answer.

5

$160y^{2}$

$26y^{2}$

$6y^{2}$

Explanation

Multiply the whole numbers and add the exponents:

$16y* 10y =160y^{2}$

Answer: $160y^{2}$

6

Mulitply:

$x^{2} -2x-35$

$x^{2} -35x-12$

$x^{2} +2x-35$

$x^{2} -35$

$x^{2} -12x-35$

Explanation

The FOIL method works nicely here:

$(x-7 )(x+5) = x \cdot x + x \cdot5 + (-7) \cdot x + (-7) \cdot 5 = x^{2} +5x -7x -35 = x^{2} - 2x -35$

7

Multiply:

Explanation

Use the distributive property:

$4x(3x-2)=(4x\times3x)-(4x\times2)=12x^2-8x$

8

$\frac{\begin{array}[b]{r}20\ \times\ 6\end{array}}{ \ \ \ \space}$

Explanation

To solve this problem, we multiply by the ones position and the tens position.

$6\times0=0$

$6\times2=12$

$\frac{\begin{array}[b]{r}20\ \times\ 6\end{array}}{ \ \space120}$

9

Define an operation on the real numbers as follows:

For all real values of and ,

$a \bigstar b = ab-a$

is a positive number. Which is the greater quantity?

(a) $N \bigstar (-N)$

(b) $(-N) \bigstar N$

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

Explanation

$a \bigstar b = ab-a$ , so

$N \bigstar (-N) = N (-N ) - N= -N^{2} - N$

$(-N) \bigstar N = (-N )N - (-N)= -N^{2}+N$

Since is positive,

,

and by the addition property of inequality,

$-N^{2}+N > -N^{2} - N$

and

$(-N) \bigstar N > N \bigstar (-N)$

10

Which quantity is greater?

$(a)\ -4(x^{2}+2x-10)$

$(b)\ -4x^2-8x-40$

is greater

is greater

and are equal

It is not possible to tell from the information given

Explanation

$(a)= -4(x^{2}+2x-10)$

Use the distributive property to expand.

To compare the expressions, we can remove common factors. and are common in both expressions, so they can be removed:

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