ISEE Upper Level Quantitative Reasoning - How to subtract exponents

Define as follows:

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

Evaluate .

Explanation

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

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

Simplify the expresseion:

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

The expression is already simplified.

$x^{4}$

Explanation

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.

Subtract and simplify:

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

$18x^{2}-3x-26$

$4x^{2}-3x-26$

$18x^{2}-13x-26$

$4x^{2}-3x+12$

$18x^{2}-3x+12$

Explanation

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:

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

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

Evaluate

Explanation

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

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

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

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

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

$\frac{1}{2}$

Undefined

Explanation

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

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

Evaluate and separately:

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

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

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

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

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

Define a function as follows:

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

Evaluate .

Explanation

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

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

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

Define an operation $\Upsilon$ as follows:

For all real numbers ,

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

Evaluate: $0.2; \Upsilon; 0.1$ .

Explanation

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

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

Subtract and simplify:

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

$2x^{2}+ 5x-16$

$2x^{2}-2.6x+2$

$2x^{2}+ 5x+2$

$2x^{2}-2.6x-16$

The correct answer is not among the other responses.

Explanation

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:

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

Define an operation $\Upsilon$ as follows:

For all real numbers ,

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

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

$\frac{3}{64}$

$\frac{1}{32}$

$-\frac{1}{48}$

$\frac{1}{16}$

Explanation

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

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

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

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

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

$= \frac{3}{64}$

Define as follows:

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

Evaluate .

Explanation

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

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

How to subtract exponents

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