Problem 260 In the following exercises, simp... [FREE SOLUTION]

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Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 260

In the following exercises, simplify. $$\frac{a^{3} \cdot a^{4}}{a^{7}}$$

Short Answer

Expert verified

Step by step solution

Multiply the exponents in the numerator

The given expression is $ \frac{a^3 \times a^4}{a^7} $. First, use the product of powers property, which states $ a^m \times a^n = a^{m+n} $, to combine the exponents in the numerator. $ a^3 \times a^4 = a^{3+4} = a^7 $. This simplifies the expression to $ \frac{a^7}{a^7} $.

Simplify the fraction

Now, use the quotient of powers property, which states $ \frac{a^m}{a^n} = a^{m-n} $, to simplify $ \frac{a^7}{a^7} $. This results in $ a^{7-7} = a^0 $.

Evaluate the result

Any non-zero number raised to the power of zero equals 1. Hence, $ a^0 = 1 $.

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Key Concepts

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

product of powers property

When we multiply two expressions that have the same base, we use the product of powers property. This property helps us simplify expressions by adding the exponents.

For instance, in the expression $a^3 \times a^4$, both terms have the same base, which is $a$. According to the product of powers property, we add the exponents: $3 + 4$. So, $a^3 \times a^4 = a^{7}$.

This is very useful when simplifying complex expressions. Just remember: same base, add the exponents. The result has the same base but with a new exponent that is the sum of the original exponents.

quotient of powers property

The quotient of powers property is the opposite of the product of powers property. When dividing expressions with the same base, we subtract the exponents.

Consider our intermediate expression \frac{a^7}{a^7}\. Here, both the numerator and the denominator have the same base $a$. According to the quotient of powers property, we subtract the exponents of the numerator and the denominator: \[a^{7-7}\].

This simplifies to $a^0$, since \[7 - 7 = 0\]. Remember: same base, subtract the exponents. The new exponent is simply the result of the subtraction.

zero exponent rule

The zero exponent rule states that any non-zero number raised to the power of zero is equal to 1. This might seem a bit tricky, but it's easy once you remember the property.

Let's look at our simplified expression from earlier, $a^0$. According to the zero exponent rule, any base $a$, as long as it's not zero, raised to the power of zero becomes 1. Thus, $a^0 = 1$.

This rule helps us simplify expressions significantly, rendering complex calculations a whole lot easier once we spot an exponent of zero.

So remember: any number (except zero) raised to the power of zero equals 1.

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91影视

In the following exercises, simplify. $$\frac{a^{3} \cdot a^{4}}{a^{7}}$$

Short Answer

Step by step solution

Multiply the exponents in the numerator

Simplify the fraction

Evaluate the result

Key Concepts

product of powers property

quotient of powers property

zero exponent rule

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