How to subtract exponents - ISEE Upper Level Quantitative Reasoning

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Question

Simplify:

Answer

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

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Question

Evaluate:

Answer

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

Now they must be multiplied out before they can be added:

$=5\times 5- 3\times 5\times 5\times 5-4\times 7\times 7$

$=25-3\times 125-4\times 49$

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Question

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $0.2; \Upsilon; 0.1$ .

Answer

$a \Upsilon b = a^{4} - b^{3}$

$0.2; \Upsilon; 0.1= 0.2^{4} - 0.1^{3}$

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Question

Subtract and simplify:

$\left ( 11 x^{2}- 8x-7 \right )- \left (-7 x^{2}-5x+19\right )$

Answer

Consider a vertical subtraction process:

$\begin{matrix} 11 x^{2}- 8x-7 \\underline{ - (-7 x^{2}-5x+19)} \end{matrix}$

Rewrite as the addition of the opposite of the second expression, as follows:

$\begin{matrix} 11 x^{2}- 8x-7 \\underline{ ; ; 7 x^{2}+5x-19 } \ 18x^{2}-3x-26 \end{matrix}$

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Question

Subtract and simplify:

$\left ( 4x^{2}+ 1.2x-7 \right )- \left ( 2x^{2}-3.8x+9\right )$

Answer

Consider a vertical subtraction process:

$\begin{matrix} 4x^{2}+ 1.2x-7\\underline{ - (2x^{2}-3.8x+9)} \end{matrix}$

Rewrite as the addition of the opposite of the second expression, as follows:

$\begin{matrix} ; ; 4x^{2}+ 1.2x-7\\underline{ -2x^{2}+3.8x-9}\ ; 2x^{2}+; ; 5x-16\end{matrix}$

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Question

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $\frac{1}{2} \Upsilon \frac{1}{4}$ .

Answer

$a \Upsilon b = a^{4} - b^{3}$ , so

$\frac{1}{2} \Upsilon \frac{1}{4} = \left ( \frac{1}{2} \right )^{4} - \left (\frac{1}{4} \right )^{3}$

$= \frac{1}{2^{4}} - \frac{1}{4^{3}}$

$= \frac{1}{16} - \frac{1}{64}$

$= \frac{4}{64} - \frac{1}{64}$

$= \frac{3}{64}$

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Question

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $(-4) \Upsilon (-3)$

Answer

$a \Upsilon b = a^{4} - b^{3}$

$(-4) \Upsilon (-3) = (-4)^{4}- (-3) ^{3}$

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Question

Define as follows:

$f(x) = x^{5} - x^{3}$

Evaluate .

Answer

$f(x) = x^{5} - x^{3}$

$f(-4) = (-4)^{5} - (-4)^{3}= -1,024- (-64) = -960$

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Question

Simplify the expresseion:

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

Answer

Variable terms can be combined by adding and subtracting if and only of they are like - that is, if each exponent of each variable is the same. In the given expression, no two exponents are the same. The terms cannot be combined, and the expression is already simplified.

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Question

Define a function as follows:

$g(x)=\left | x^{5}-x^{2}\right |$

Evaluate .

Answer

$g(x)=\left | x^{5}-x^{2}\right |$

$g(3)=\left | 3^{5}-3^{2}\right | = \left | 243-9\right | = \left | 234\right | = 234$

$g(-3)=\left | \left ( -3\right )^{5}- \left ( -3\right )^{2}\right | = \left | -243- 9\right | = \left | -252\right | = 252$

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Question

Define a function $g(x)=\left | x^{5}-x^{3}\right |$ .

Evaluate

Answer

$g(x)=\left | x^{5}-x^{3}\right |$

$g(4)=\left | 4^{5}-4^{3}\right | = \left | 1,024-64\right | = \left | 960\right |= 960$

$g(4)=\left | \left (-4 \right )^{5}- \left (-4 \right )^{3}\right | = \left | -1,024-\left ( - 64 \right )\right | = \left | -960\right |= 960$

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Question

Define $f(x)= x^{6} + x^{3}$ and $g(x) = x^{6}-x^{3}$ .

Evaluate $\left (\frac{f}{g} \right ) (-1)$ .

Answer

$\left (\frac{f}{g} \right ) (x) = \frac{f(x)}{g(x)}$ , by definition, so

$\left (\frac{f}{g} \right ) (-1)= \frac{f(-1)}{g(-1)}$

Evaluate and separately:

$f(x)= x^{6} + x^{3}$

$f(-1)= (-1)^{6} + (-1)^{3} = 1+(-1)= 0$

$g(x) = x^{6}-x^{3}$

$g(-1)= (-1)^{6} - (-1)^{3} = 1-(-1)= 2$

$\left (\frac{f}{g} \right ) (-1)= \frac{f(-1)}{g(-1)} = \frac{0}{2}= 0$

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Question

Define as follows:

$f(x) = x^{6} - x^{5}$

Evaluate .

Answer

$f(x) = x^{6} - x^{5}$

$f(x) = (-3)^{6} - (-3)^{5} = 729-(-243)= 972$

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Question

Simplify:

Answer

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

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Question

Evaluate:

Answer

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

Now they must be multiplied out before they can be added:

$=5\times 5- 3\times 5\times 5\times 5-4\times 7\times 7$

$=25-3\times 125-4\times 49$

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Question

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $0.2; \Upsilon; 0.1$ .

Answer

$a \Upsilon b = a^{4} - b^{3}$

$0.2; \Upsilon; 0.1= 0.2^{4} - 0.1^{3}$

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Question

Subtract and simplify:

$\left ( 11 x^{2}- 8x-7 \right )- \left (-7 x^{2}-5x+19\right )$

Answer

Consider a vertical subtraction process:

$\begin{matrix} 11 x^{2}- 8x-7 \\underline{ - (-7 x^{2}-5x+19)} \end{matrix}$

Rewrite as the addition of the opposite of the second expression, as follows:

$\begin{matrix} 11 x^{2}- 8x-7 \\underline{ ; ; 7 x^{2}+5x-19 } \ 18x^{2}-3x-26 \end{matrix}$

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Question

Subtract and simplify:

$\left ( 4x^{2}+ 1.2x-7 \right )- \left ( 2x^{2}-3.8x+9\right )$

Answer

Consider a vertical subtraction process:

$\begin{matrix} 4x^{2}+ 1.2x-7\\underline{ - (2x^{2}-3.8x+9)} \end{matrix}$

Rewrite as the addition of the opposite of the second expression, as follows:

$\begin{matrix} ; ; 4x^{2}+ 1.2x-7\\underline{ -2x^{2}+3.8x-9}\ ; 2x^{2}+; ; 5x-16\end{matrix}$

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Question

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $\frac{1}{2} \Upsilon \frac{1}{4}$ .

Answer

$a \Upsilon b = a^{4} - b^{3}$ , so

$\frac{1}{2} \Upsilon \frac{1}{4} = \left ( \frac{1}{2} \right )^{4} - \left (\frac{1}{4} \right )^{3}$

$= \frac{1}{2^{4}} - \frac{1}{4^{3}}$

$= \frac{1}{16} - \frac{1}{64}$

$= \frac{4}{64} - \frac{1}{64}$

$= \frac{3}{64}$

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Question

Define an operation $\Upsilon$ as follows:

For all real numbers ,

$a \Upsilon b = a^{4} - b^{3}$ .

Evaluate: $(-4) \Upsilon (-3)$

Answer

$a \Upsilon b = a^{4} - b^{3}$

$(-4) \Upsilon (-3) = (-4)^{4}- (-3) ^{3}$

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