ISEE Upper Level Quantitative Reasoning - How to add exponents

What is the value of the expression:

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

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

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

$x^{6}\cdotx^{y+2}$

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

Explanation

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

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

Simplify the expression:

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

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

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

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

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

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

Explanation

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,

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

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

Simplify:

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

$6x^{6}$

$6x^{8}$

$5x^{6}$

$6x^{2}$

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

Explanation

When multiplying exponents, the exponents are added together.

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

Simplify:

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

Explanation

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:

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

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

$\frac{1}{27}$

$\frac{1}{3^{13}}$

$3^{13}$

$\frac{1}{9}$

Explanation

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: $\frac{1}{27}$

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

$2x^{10}$

$x^{10}$

$2x^{21}$

$x^{21}$

Explanation

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:

$2x^{10}$

Simplify:

Explanation

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

Which of the following is equivalent to the expression below?

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

$2^{2x+5}$

$8^{5x}$

$6^{5x}$

$8^{10x}$

Explanation

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:

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

$2^{2x+5}$

Simplify:

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

$x^{10}$

$x^{24}$

$x^{2}$

$x^{-2}$

$x^{-10}$

Explanation

When multiplying exponents, the exponents are added together.

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

Evaluate:

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

Explanation

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:

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

How to add exponents

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