Numbers and Operations - ISEE Upper Level Mathematics Achievement

Card 0 of 20

Question

Simplify the expression:

$6x^{4} + 8x^{3} - 4x^{2}$

Answer

$6x^{4} + 8x^{3} - 4x^{2}$

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

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Question

Simplify:

Answer

In order to add exponential terms, both the base and the exponent must be the same. So we can write:

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Question

Evaluate:

$\frac{x^0+y^0+1}{x^0-y^0-1}$

Answer

Based on the zero-exponent rule we have:

That means any non-zero number raised to the zero power is equal to . So we can write:

$\frac{1+1+1}{1-1-1}=\frac{3}{-1}=-3$

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Question

Which of the following is equivalent to the expression below?

$4^{x}\cdot2^{5}$

Answer

When exponents are multiplied by one another, and the base is the same, the exponents can be added together.

The first step is to try to create a common base.

$4^{x}\cdot2^{5}$

Given that the square of 2 is for, the expression can be rewritten as:

$2^{2}^{x}\cdot 2^{5}$

$2^{2x+5}$

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Question

What is the value of the expression:

$x^{3}\cdotx^{y}\cdot x^{2+y}$

Answer

When values, having the same base, are multiplied by one another, the exponents are added together and the base stays the same.

Thus,

$x^{3}\cdotx^{y}\cdot x^{2+y}$

is equal to

$x^{4}\cdotx^{y+2}$

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Question

Simplify: $x^{7}\cdot2x^{3}$

Answer

When multiplying exponents, the exponents are added together. Thus, 3 and 7 are added together for a sum of 10. In this problem, the "2" becomes a coefficient in front of the x. Therefore, the correct answer is:

$2x^{10}$

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Question

Simplify the expression:

$3x^{4}+6x^{3}-9x^{2}$

Answer

To simplify this problem we need to factor out a

$3x^{4}+6x^{3}-9x^{2}$

$=3x^{2}(x^{2}+2x-3)$

We can do this because multiplying exponents is the same as adding them. Therefore,

$=3x^{2}(x^{2}+2x-3)$

$=3x^2 \cdot x^2 +3x^2\cdot2x -3x^2\cdot3=3x^{2+2}+3\cdot2x^{2+1}-3\cdot 3x^2$

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Question

Simplify:

$x^{6}\cdot x^{4}$

Answer

When multiplying exponents, the exponents are added together.

$x^{6}\cdot x^{4} = x^{6+4}=x^{10}$

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Question

Simplify:

$2x^{2}\cdot 3x^{4}$

Answer

When multiplying exponents, the exponents are added together.

$2x^{2}\cdot 3x^{4}=6x^{2+4}=6x^{6}$

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Question

Solve: $3^5 \cdot 3^{-8}$

Answer

It is not necessary to evaluate both terms and multiply.

According to the rules of exponents, when we have the same bases raised to some power that are multiplied with each other, we can add the powers.

$3^5 \cdot 3^{-8} = 3^{5+( -8)} = 3^{-3}$

This term can be rewritten as a fraction.

$3^{-3 } = \frac{1}{3^3} = \frac{1}{27}$

The answer is: $\frac{1}{27}$

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Question

Simplify:

$(6a^3)(\frac{1}{3}a^2)$

Answer

Based on the product rule for exponents in order to multiply two exponential terms with the same base, add their exponents:

$(x)^a(x)^b=x^{a+b}$

So we can write:

$(6a^3)\left(\frac{1}{3}a^2\right)=\left(\frac{6}{3}\right)\times a^{3+2}=2a^5$

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Question

Simplify: $\frac{x^{50}}{x^{10}}$

Answer

Apply the quotient of powers rule.

$\frac{x^{50}}{x^{10}} = x^{50-10} = x^{40}$

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Question

Simplify: $\frac{12x^{12}}{3x^{3}}$

Answer

Separate the fraction and apply the quotient of powers rule:

$\frac{12x^{12}}{3x^{3}} = \frac{12}{3} \cdot \frac{x^{12}}{x^{3}} = 4x^{12-3} = 4x^{9}$

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Question

Evaluate:

$\frac{120}{12^{0}}$

Answer

Any nonzero number raised to the power of 0 is equal to 1, so $12^{0} = 1$ , and

$\frac{120}{12^{0}} = \frac{120}{1} = 120$

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Question

Evaluate:

$\frac{25}{10 ^{0} -5^{0}}$

Answer

Any nonzero number raised to the power of 0 is equal to 1, so $5^{0} = 10 ^{0} =1$ . Therefore,

$\frac{25}{10 ^{0} -5^{0}} = \frac{25}{1-1} = \frac{25}{0}$

However, an expression with a denominator of 0 is undefined, so that is the correct choice.

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Question

Simplify:

$\frac{6x^6}{2x^3}$

Answer

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

We should separate the fraction and apply the quotient rule for exponents:

$\frac{6x^6}{2x^3}=\frac{6}{2}\times \frac{x^6}{x^3}=3\times x^{6-3}$

$\Rightarrow \frac{6x^6}{2x^3}=3x^3$

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Question

Simplify:

$\frac{4x^8}{2x^0}$

Answer

Based on the quotient rule for exponents, we know that for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

We also know that any non-zero number raised to the power of is .

$\frac{4x^8}{2x^0}=\frac{4}{2}\times \frac{x^8}{1}=2x^8$

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Question

Simplify:

$\frac{6x^4-6}{3x^2+3x^0}$

Answer

We know that any non-zero number raised to the power of is equal to .

$\frac{6x^4-6}{3x^2+3x^0}=\frac{6(x^4-1)}{3(x^2+x^0)}=\frac{6(x^4-1)}{3(x^2+1)}$

Now factor:

$\frac{6(x^4-1)}{3(x^2+1)}=\frac{6}{3}\frac{(x^2-1)(x^2+1)}{x^2+1}=2(x^2-1)$

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Question

Evaluate:

$\frac{4^0}{2\times 5^0-12^0}$

Answer

We know that any non-zero number raised to the power of is equal to .

$\frac{4^0}{2\times 5^0-12^0}=\frac{1}{2\times 1-1}=\frac{1}{2-1}=1$

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Question

Evaluate:

$\frac{y^{-2}}{y^{-5}}$

Answer

Based on the quotient rule for exponents, we know that, for any non-zero number and any integers and ,

$\frac{x^a}{x^b}=x^{a-b}$ .

$\frac{y^{-2}}{y^{-5}}=y^{-2-(-5)}=y^{-2+5}=y^{3}$

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