Problem 91 Simplify each expression. $$\f... [FREE SOLUTION]

91影视

Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 8: Problem 91

Simplify each expression. $$\frac{\left(c^{3}\right)^{2} c^{4}}{\left(c^{-1}\right)^{3}}$$

Short Answer

Expert verified

c^{13}

Step by step solution

- Apply Power Rule

Use the power rule \[ (a^m)^n = a^{m \times n} \] to simplify the terms within the numerator and the denominator. \[ (c^3)^2 = c^{3 \times 2} = c^6 \] and \[ (c^{-1})^3 = c^{-1 \times 3} = c^{-3} \]

- Combine Exponents in the Numerator

Combine the exponents in the numerator \[ c^6 \times c^4 = c^{6 + 4} = c^{10} \]

- Simplify the Fraction

Now, write the expression as a single fraction. \[ \frac{c^{10}}{c^{-3}} \] Use the division rule for exponents \[\frac{a^m}{a^n} = a^{m-n} \], which gives us \[ c^{10 - (-3)} = c^{10 + 3} = c^{13} \]

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Key Concepts

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

power rule

Let's begin with the power rule. This rule helps us handle expressions where exponents are raised to another exponent. The power rule can be stated as:

\[ (a^m)^n = a^{m \times n} \]

Here, if you have an expression like $ (c^3)^2 $, you use the power rule to simplify it as follows:

\[ (c^3)^2 = c^{3 \times 2} = c^6 \]

Next, apply the same rule to the denominator's term $ (c^{-1})^3 $:

\[ (c^{-1})^3 = c^{-1 \times 3} = c^{-3} \]

combining exponents

Now that we have simplified the individual terms in our expression, let's move to combining exponents in the numerator. When you multiply two terms with the same base, you simply add their exponents. This is known as the product rule for exponents, and it can be stated as:

\[ a^m \times a^n = a^{m+n} \]

Applying this rule to the expression $ c^6 \times c^4 $, we get:

\[ c^6 \times c^4 = c^{6+4} = c^{10} \]

division rule

After combining the exponents in the numerator, it's time to fraction to a single expression. The key rule here is the quotient rule for exponents. This rule is applied when you divide terms with the same base. The rule can be summarized as:

\[ \frac{a^m}{a^n} = a^{m-n} \]

We will apply this rule to our simplified fraction $ \frac{c^{10}}{c^{-3}} $:

\[ \frac{c^{10}}{c^{-3}} = c^{10-(-3)} = c^{10+3} = c^{13} \]

simplifying fractions

Simplifying fractions involves reducing them to the simplest form. In our exercise, we simplified the fraction using the quotient rule for exponents. To recap:

We used the power rule to handle exponents raised to another exponent.
We combined exponents in the numerator using the product rule.
We applied the quotient rule to simplify the fraction.

Putting it all together, we simplified the original expression $ \frac{(c^3)^2 c^4}{(c^{-1})^3} $ to $ c^{13} $. Understanding and practicing these rules will make future problems easier!

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91影视

Simplify each expression. $$\frac{\left(c^{3}\right)^{2} c^{4}}{\left(c^{-1}\right)^{3}}$$

Short Answer

Step by step solution

- Apply Power Rule

- Combine Exponents in the Numerator

- Simplify the Fraction

Key Concepts

power rule

combining exponents

division rule

simplifying fractions

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