Simplifying Exponents

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Simplify the following expression:

$13x^{24}$

$40x^{24}$

Explanation

Simplify the following expression:

When multiplying exponents, we need to recall the rule. When we multiply exponents of like base together, we add the exponents. This is seperate from the numbers out in front (the 5 and the 8) those two numbers will be multiplied as normal.

We can sort of rearrange the expression to get the following:

$5x^6*8x^3=(5*8)(x^6*x^3)=40x^{6+3}=40x^9$

Making our answer:

Simplify the following expression.

$\frac{25b^4}{5b^2}$

Explanation

Simplify the following expression

$\frac{25b^4}{5b^2}$

To simplify this, we need to subtract our exponents and divide our whole numbers.

When we do this, we get the following.

$\frac{25b^4}{5b^2}=\frac{25}{5}\frac{b^4}{b^2}=\frac{5\cdot 5b^2\cdot b^2}{5b^2}=5b^2$

Simplify.

$x^{15}*x^{17}$

$x^{32}$

$x^{22}$

$x^{255}$

$x^{285}$

$x^{-2}$

Explanation

When multiplying exponents with the same base, you just have to add the exponents.

$x^{15}*x^{17}=x^{15+17}=x^{32}$

Multiply: $8^{\frac{2}{3}} \times 8^{\frac{3}{2}}$

$64\sqrt2$

$16\sqrt2$

$32\sqrt2$

Explanation

The bases of the exponents are common. This means we can add the fractions.

$8^{\frac{2}{3}} \times 8^{\frac{3}{2}} = 8^{\frac{2}{3}+ \frac{3}{2}}$

The least common denominator is six.

$\frac{2}{3}+ \frac{3}{2} = \frac{4}{6}+ \frac{9}{6} = \frac{13}{6}$

This becomes the power of the exponent.

$8^{\frac{13}{6}}$

Break up the fraction in terms so that each can be reduced.

$8^{\frac{13}{6}} = 8^{\frac{12}{6}} \cdot 8^{\frac{1}{6}} = 8^2 \cdot 8^{\frac{1}{6}}$

Since we do not know term $8^{\frac{1}{6}}$ , it can be rewritten in base two, and .

Rewrite this term as a replacement of $8^{\frac{1}{6}}$ , and multiply the power of the exponent in base two with the power of the exponent in base eight.

$8^2\cdot 8^{\frac{1}{6}} = 8^2 \cdot (2^3)^{(\frac{1}{6})} = 8^2\cdot 2^{\frac{1}{2}}$

Simplify the terms. A value to the power of one-half is the square root of that number.

$8^2\cdot 2^{\frac{1}{2}} = 64\sqrt2$

The answer is: $64\sqrt2$

Simplify.

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

$x^{1.25}$

$x^{20}$

$x^{-1}$

Explanation

When dividing exponents, we must first check to see if the factors have the same base. If they have the same bases, then we can subtract the exponents.

$\frac{x^5}{x^4}=x^{5-4}$

$x^{5-4}=x^{1}=x$

Simplify:

$\frac{(x^2y^3)^9}{xy^7}$

$x^{17}y^{20}$

$x^{10}y^5$

$\frac{x^9}{y^{36}}$

$x^{11}y^{27}$

Explanation

Recall that when an exponent is raised to another exponent, you will need to multiply the two exponents together.

Start by simplifying the numerator:

$(x^2y^3)^9=x^{18}y^{27}$

Now, place this on top of the denominator and simplify. Recall that when you divide exponents that have the same base, you will subtract the exponent in the denominator from the exponent in the numerator.

$\frac{x^{18}y^{27}}{xy^7}=x^{17}y^{20}$

Evaluate:

$x^{35}$

$x^{12}$

$x^{24}$

$x^{57}$

$x^{36}$

Explanation

When an exponent is being raised by another exponent, we just multiply the powers and keep the base the same.

$(x^5)^7=x^{5&*7}=x^{35}$

Simplify.

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

$x^{1.25}$

$x^{20}$

$x^{-1}$

Explanation

When dividing exponents, we must first check to see if the factors have the same base. If they have the same bases, then we can subtract the exponents.

$\frac{x^5}{x^4}=x^{5-4}$

$x^{5-4}=x^{1}=x$

Simplify:

$\frac{6^{12}}{6^{18}}$

$6^{-6}$

$6^{30}$

$6^{15}$

$-6^{6}$

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

Explanation

When dividing exponents with the same base, we just subtract the exponents.

$\frac{6^{12}}{6^{18}}=6^{12-18}=6^{-6}$

Multiply: $8^{\frac{2}{3}} \times 8^{\frac{3}{2}}$

$64\sqrt2$

$16\sqrt2$

$32\sqrt2$

Explanation

The bases of the exponents are common. This means we can add the fractions.

$8^{\frac{2}{3}} \times 8^{\frac{3}{2}} = 8^{\frac{2}{3}+ \frac{3}{2}}$

The least common denominator is six.

$\frac{2}{3}+ \frac{3}{2} = \frac{4}{6}+ \frac{9}{6} = \frac{13}{6}$

This becomes the power of the exponent.

$8^{\frac{13}{6}}$

Break up the fraction in terms so that each can be reduced.

$8^{\frac{13}{6}} = 8^{\frac{12}{6}} \cdot 8^{\frac{1}{6}} = 8^2 \cdot 8^{\frac{1}{6}}$

Since we do not know term $8^{\frac{1}{6}}$ , it can be rewritten in base two, and .

Rewrite this term as a replacement of $8^{\frac{1}{6}}$ , and multiply the power of the exponent in base two with the power of the exponent in base eight.

$8^2\cdot 8^{\frac{1}{6}} = 8^2 \cdot (2^3)^{(\frac{1}{6})} = 8^2\cdot 2^{\frac{1}{2}}$

Simplify the terms. A value to the power of one-half is the square root of that number.

$8^2\cdot 2^{\frac{1}{2}} = 64\sqrt2$

The answer is: $64\sqrt2$

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