Problem 579 Simplify. $$\left(\frac{p^{4} ... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 579

Simplify. $$\left(\frac{p^{4} \cdot p^{5}}{p^{3}}\right)^{2}$$

Short Answer

Expert verified

The simplified form is $p^{12}$.

Step by step solution

Combine the Exponents in the Numerator

Use the property of exponents, $a^m \times a^n = a^{m+n}$, to combine the exponents in the numerator. \[\frac{p^{4} \times p^{5}}{p^{3}} = \frac{p^{4+5}}{p^{3}} = \frac{p^{9}}{p^{3}}\]

Subtract the Exponents in the Fraction

Use the property of exponents $\frac{a^m}{a^n} = a^{m-n}$ to simplify the fraction. \[\frac{p^{9}}{p^{3}} = p^{9-3} = p^{6}\]

Apply the Exponent Outside the Parentheses

Now raise the expression to the power of 2 using the property $(a^m)^n = a^{m \times n}$. \[(p^{6})^2 = p^{6 \times 2} = p^{12}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Exponents

When you have the same base, combining exponents involves adding them. For instance, if we have something like $a^m \times a^n$, this becomes $a^{m+n}$. This rule makes math easier because it simplifies expressions where bases are multiplied.

In the given problem, $p^4 \cdot p^5$, both terms have the same base, which is $p$. According to the rule, we can combine them into $p^{4+5}$, which simplifies to $p^9$. This step reduces the number of terms we need to deal with.

Simplifying Fractions with Exponents

When you deal with fractions that have exponents, the key idea is to subtract the exponents if the bases are the same. This is expressed as $\frac{a^m}{a^n} = a^{m-n}$.

In the problem, after combining the exponents in the numerator to get $p^9$, we have the fraction $\frac{p^9}{p^3}$. Applying the rule, subtract the exponent of the denominator from the numerator: $p^{9-3} = p^6$. This step simplifies the fraction significantly, making our expression much easier to manage.

Power of a Power Property

The power of a power property states that if you have an exponent raised to another exponent, you multiply the exponents. This is written as $(a^m)^n = a^{m \times n}$.

In our problem, we ended with $(p^6)^2$. To simplify this, we multiply the exponents: $p^{6 \times 2} = p^{12}$. This rule helps us efficiently deal with nested exponents and further simplifies the expression.

Overall, by following these steps and applying the properties correctly, we simplified $\left(\frac{p^4 \cdot p^5}{p^3}\right)^2$ to $p^{12}$.

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91影视

Simplify. $$\left(\frac{p^{4} \cdot p^{5}}{p^{3}}\right)^{2}$$

Short Answer

Step by step solution

Combine the Exponents in the Numerator

Subtract the Exponents in the Fraction

Apply the Exponent Outside the Parentheses

Key Concepts

Combining Exponents

Simplifying Fractions with Exponents

Power of a Power Property

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