How to add exponents - ISEE Upper Level Quantitative Reasoning

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Question

Two quantities are given - one in Column A and the other in Column B. Compare the quantities in the two columns.

Assume, in both columns, that .

Column A Column B

Answer

When you are adding and subtracting terms with exponents, you combine like terms. Since both columns have expressions with the same exponent throughout, you are good to just look at the coefficients. Remember, a coefficient is the number in front of a variable. Therefore, Column A is since . Column B is since . We can see that Column B is greater.

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Question

Which is the greater quantity?

(A) The sum of the first ten perfect square integers

(B) The sum of the first five perfect cube integers

Answer

The sum of the first ten perfect square integers:

$\begin{matrix} ;; ; 1\ ;; ; 4\ ;; ; 9\ ;16\ ;25\ ;36\ ;49\ ;64\ ;81\ \underline{100}\ 385 \end{matrix}$

The sum of the first five perfect cube integers:

$\begin{matrix} ;; ; 1\ ;; ; 8\ ;27\ ;64\ \underline{125}\ 225 \end{matrix}$

(A) is greater.

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Question

Add all of the perfect squares between 50 and 100 inclusive.

Answer

The perfect squares between 50 and 100 inclusive are

$8^{2} = 64$

$9^{2} = 81$

$10 ^{2} = 100$

Their sum is

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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

Two quantities are given - one in Column A and the other in Column B. Compare the quantities in the two columns.

Assume, in both columns, that .

Column A Column B

Answer

When you are adding and subtracting terms with exponents, you combine like terms. Since both columns have expressions with the same exponent throughout, you are good to just look at the coefficients. Remember, a coefficient is the number in front of a variable. Therefore, Column A is since . Column B is since . We can see that Column B is greater.

Compare your answer with the correct one above

Question

Which is the greater quantity?

(A) The sum of the first ten perfect square integers

(B) The sum of the first five perfect cube integers

Answer

The sum of the first ten perfect square integers:

$\begin{matrix} ;; ; 1\ ;; ; 4\ ;; ; 9\ ;16\ ;25\ ;36\ ;49\ ;64\ ;81\ \underline{100}\ 385 \end{matrix}$

The sum of the first five perfect cube integers:

$\begin{matrix} ;; ; 1\ ;; ; 8\ ;27\ ;64\ \underline{125}\ 225 \end{matrix}$

(A) is greater.

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Question

Add all of the perfect squares between 50 and 100 inclusive.

Answer

The perfect squares between 50 and 100 inclusive are

$8^{2} = 64$

$9^{2} = 81$

$10 ^{2} = 100$

Their sum is

Compare your answer with the correct one above

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:

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

Compare your answer with the correct one above

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