How to multiply exponents - ISEE Upper Level: Quantitative Reasoning

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Question

$\dpi{100} (7-4)^{3}=$

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Answer

Subtract the numbers inside the parentheses first. This leaves you with 3, which you then raise to the 3rd power:

$\dpi{100} 3\times 3\times 3=27$

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Question

Evaluate:

$\frac{1^{0}+2 ^{0}+3 ^{0}}{6^{0}}$

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Answer

Any nonzero number taken to the power of 0 is equal to 1, so

$\frac{1^{0}+2 ^{0}+3 ^{0}}{6^{0}} = \frac{1+1+1}{1} = \frac{3}{1} = 3$

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Question

Multiply:

$(4r^{2} - 7) (5r^{2}+ 8 )$

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Answer

Use the FOIL method:

$(4r^{2} - 7) (5r^{2}+ 8 )$

$=4r^{2} \cdot 5r^{2} + 4r^{2} \cdot 8 -7 \cdot 5r^{2} -7 \cdot 8$

$=4\cdot 5 \cdot r^{2}\cdot r^{2} + 4\cdot 8\cdot r^{2} -7 \cdot 5\cdot r^{2} -7 \cdot 8$

$=20 \cdot r^{2+2} + 32r^{2} -35r^{2} -56$

$=20r^{4} +\left ( 32-35 \right )r^{2} -56$

$=20r^{4} -3r^{2} -56$

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Question

Simplify:

$(x^{2\sqrt{2}})^{3\sqrt{2}}$

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Answer

Based on the power rule, we know that in order to raise a power to a power we need to multiply the exponents, i.e.

$(x^{m})^{n}=x^{m\times n}$ .

$(x^{2\sqrt{2}})^{3\sqrt{2}}=x^{2\sqrt{2}\times 3\sqrt{2}}=x^{2\times 3\times 2}=x^{12}$

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Question

Simplify:

$(\frac{12x^{3}}{3x^{-1}})^{4}$

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Answer

The Negative Exponent Rule says $a^{-n}=\frac{1}{a^n}$ .

$(\frac{12x^{3}}{3x^{-1}})^{4}=(\frac{12}{3}\times x^{3+1})^{4}=(4x^4)^{4}$

The power rule says that, in order to raise a power to a power, we need to multiply the exponents, i.e. $(x^{m})^{n}=x^{m\times n}$ .

$(\frac{12x^{3}}{3x^{-1}})^{4}=(4x^4)^{4}=4^4\times x^{4\times 4}=256x^{16}$

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Question

Simplify:

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

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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

Evaluate:

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

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Answer

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. In addition, based on the product rule for exponents in order to multiply two exponential terms with the same base, we need to add their exponents:

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

So we can write:

$-x(x^3)^{2}=-x(x^{3\times 2})=-x(x^6)=-x^{1+6}=-x^7$

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Question

Evaluate:

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

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Answer

Based on the negative exponent rule we have:

$a^{-n}=\frac{1}{a^n}$

which says negative exponents in the numerator get moved to the denominator and become positive exponents. And negative exponents in the denominator get moved to the numerator and become positive exponents. So we can write:

$-2x^2(x^{-3})^{3}=-2x^2\left(\frac{1}{x^3}\right)^{3}$

In addition, based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. We also know that when a fraction is raised to a power, the numerator and the denominator are both raised to that power. So we can write:

$-2x^2(x^{-3})^{3}=-2x^2\left(\frac{1}{x^3}\right)^{3}=-2x^2\times \frac{1^3}{x^{3\times 3}}$

$=-2x^2\times \frac{1}{x^9}$

$=\frac{-2x^2}{x^9}$

$=-2x^{2-9}$

$=-2x^{-7}$

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Question

Evaluate:

$\frac{4x^2(x^3)^{3}}{8x^7}$

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Answer

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. In addition, based on the product rule for exponents, in order to multiply two exponential terms with the same base we need to add their exponents:

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

So we can write:

$\frac{4x^2(x^3)^{3}}{8x^7}=\frac{4x^2(x^{3\times 3})}{8x^7}=\frac{4x^2\times x^9}{8x^7}=\left(\frac{4}{8}\right)\frac{x^{2+9}}{x^7}=\frac{1}{2}\cdot \frac{x^{11}}{x^7}$

in order to divide two exponents with the same base, we can keep the base and subtract the powers. So we get:

$\frac{4x^2(x^3)^{3}}{8x^7}=\frac{1}{2}\cdot \frac{x^{11}}{x^7}=\frac{1}{2}x^{11-7}=\frac{1}{2}x^4$

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Question

If , find the value of:

$\frac{3x^{\sqrt{2}}(x^{\sqrt{3}})^\sqrt{3}}{x^{\sqrt{2}}}$

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Answer

Based on the power rule for exponents we can write:

$(a^m)^{n}=a^{m\times n}$

That means; to raise a power to a power we need to multiply the exponents. So we can write:

$\frac{3x^{\sqrt{2}}(x^{\sqrt{3}})^\sqrt{3}}{x^{\sqrt{2}}}=3\times \left(\frac{x^{\sqrt{2}}}{x^{\sqrt{2}}}\right) \times (x^{\sqrt{3}})^{\sqrt{3}}=3x^{\sqrt{3}\times \sqrt{3}}=3x^3$

Substitute and we get:

$3x^3=3\times 2^3=3\times 8=24$

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Question

What is the value of this equation?

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

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Answer

When an exponent is raised to another exponent, the exponents should be multiplied toghether. This will result in:

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

$x^{6}-2^{6}$

$x^{6}-64$

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Question

Which expression is equal to $(x^{2})^{3}$ ?

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Answer

When exponent of a value is raised to another exponent, the values of the exponents are multiplied by each other.

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

2 is multiplied by 3, and so the exponent of x is 6.

$x^{6}$

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Question

What is the value of $-1^{323}$ ?

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Answer

1 raised to any exponent will always be 1.

-1 will be equal to 1 when the exponent is even and will be equal to -1 when the exponent is odd.

Given that 323 is odd, $-1^{323}$ is equal to -1.

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Question

What is the expression below equal to?

$3x^{4}*5x^{9}$

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Answer

When exponents are multiplied by each other, the powers should be added together. Meanwhile, numbers not raised to an exponent are simply multiplied by each other.

Therefore, the answer is $15x^{13}$ , because , and .

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Question

What is the value of $(2^{6})^{\frac{1}{2}}$ ?

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Answer

When one exponent is raised to another exponent, the values of the exponents should be multiplied together. Thus,

$(2^{6})^{\frac{1}{2}}$ can be simplified to $2^{3}$ , given that $6\cdot \frac{1}{2}=3$ .

$2^{3}=2\cdot2\cdot2=8$

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Question

Simplify the following:

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Answer

Simplify the following:

Let's begin by recalling two rules

When multiplying variables with a common base, add the exponents.

When multiplying variables with a common base, multiply the coefficients.

$6x^55x^63x^2=(563)x^{5+6+2}=90x^{13}$

So, our answer is

$90x^{13}$

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Question

Simplify: $\left (y ^{3} \right )^{4}$

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Answer

Apply the power of a power property:

$\left (y ^{3} \right )^{4} = y ^{3 \cdot 4} = y^{12}$

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Question

Simplify the expression: $\left (5 a^{3}y ^{4} \right )^{2}$

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Answer

Apply the power of a product rule, then apply the power of a power rule:

$\left (5 a^{3}y ^{4} \right )^{2}$

$= 5^{2} \cdot \left (a^{3}\right )^{2} \cdot \left ( y ^{4} \right )^{2}$

$= 25 \cdot a^{ 3 \cdot 2 } \cdot y ^{ 4 \cdot 2 }$

$= 25 a^{ 6} y ^{ 8 }$

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Question

Which of the following expressions is equal to $1,234^{0}$ ?

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Answer

Any nonzero number raised to the power of 0 is equal to 1.

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Question

Which quantity is greater?

(a) $(-5) ^{3}$

(b) $(-5) ^{-3}$

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Answer

(a) $(-5) ^{3} = (-5) \cdot (-5) \cdot (-5) = 25 \cdot (-5) = -125$

(b) $(-5) ^{-3} = \frac{1}{(-5) ^{3}} = \frac{1}{-125} =- \frac{1}{125}$

(b) is the greater quantity.

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