Problem 68
Question
In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{1}{a^{-3}} $$
Step-by-Step Solution
Verified Answer
The quotient \( \frac{1}{a^{-3}} \) is rewritten as \( a^3 \).
1Step 1: Rewrite the expression with positive exponents
To eliminate negative exponents in a fraction, we can rewrite the expression using the property that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( a^{-3} \) can be rewritten as \( \frac{1}{a^3} \). This turns the original expression \( \frac{1}{a^{-3}} \) into \( \frac{1}{\frac{1}{a^3}} \).
2Step 2: Simplify the complex fraction
To simplify \( \frac{1}{\frac{1}{a^3}} \), we apply the rule of dividing by a fraction, which is equivalent to multiplying by its reciprocal. Thus, \( \frac{1}{\frac{1}{a^3}} = 1 \cdot a^3 = a^3 \).
3Step 3: Express the quotient as a product
The expression \( a^3 \) represents the product of three factors of \( a \). Thus, the original quotient \( \frac{1}{a^{-3}} \) is rewritten as \( a^3 \), which is a product without a denominator.
Key Concepts
Negative ExponentsFraction SimplificationReciprocal Rule
Negative Exponents
Negative exponents can be puzzling at first, but they are quite manageable once you understand their fundamental meaning. A negative exponent indicates that the base, which is the number or variable with the exponent, should be taken as the reciprocal. For example, with the expression \( a^{-3} \), the negative sign on the exponent signifies we should take the reciprocal of the base raised to the positive equivalent of that exponent.
This means that \( a^{-3} \) is equivalent to \( \frac{1}{a^3} \). The act of taking the reciprocal "flips" the base, which turns the exponent into a positive one. This is a handy property because it allows us to eliminate negative exponents entirely by rewriting them as fractions. Remember, turning a negative exponent into a fraction simplifies many algebraic expressions.
This means that \( a^{-3} \) is equivalent to \( \frac{1}{a^3} \). The act of taking the reciprocal "flips" the base, which turns the exponent into a positive one. This is a handy property because it allows us to eliminate negative exponents entirely by rewriting them as fractions. Remember, turning a negative exponent into a fraction simplifies many algebraic expressions.
Fraction Simplification
When dealing with complex fractions, especially those with exponents, simplification is key to cracking the problem. The main rule to remember is that dividing by a fraction is the same as multiplying by its reciprocal.
Consider the fraction \( \frac{1}{\frac{1}{a^3}} \). Simplifying this means identifying that you are dividing by \( \frac{1}{a^3} \). To simplify it, you multiply by its reciprocal. The reciprocal of \( \frac{1}{a^3} \) is \( a^3 \), so:
In this way, complex fractions become simple, and the expression becomes easier to understand and use. Focus on switching the division to multiplication by the reciprocal to smooth out these fractions efficiently.
Consider the fraction \( \frac{1}{\frac{1}{a^3}} \). Simplifying this means identifying that you are dividing by \( \frac{1}{a^3} \). To simplify it, you multiply by its reciprocal. The reciprocal of \( \frac{1}{a^3} \) is \( a^3 \), so:
- \( \frac{1}{\frac{1}{a^3}} = 1 \cdot a^3 = a^3 \)
In this way, complex fractions become simple, and the expression becomes easier to understand and use. Focus on switching the division to multiplication by the reciprocal to smooth out these fractions efficiently.
Reciprocal Rule
The reciprocal rule is a fundamental concept that is especially useful when working with fractions and exponents. A reciprocal is simply what you multiply a number by to get one. For example, the reciprocal of \( a \) is \( \frac{1}{a} \), because \( a \times \frac{1}{a} = 1 \).
This idea is crucial for simplifying expressions like \( \frac{1}{a^{-3}} \). In this expression, \( a^{-3} \) becomes \( \frac{1}{a^3} \) because of the negative exponent rule. Now, you have \( \frac{1}{\frac{1}{a^3}} \), which is a complex fraction. By understanding the reciprocal rule, you know that dividing by \( \frac{1}{a^3} \) is the same as multiplying by \( a^3 \), helping us simplify the expression to just \( a^3 \). This transformation is pivotal as it turns convoluted algebra into straightforward arithmetic steps, aiding in a deeper appreciation of the elegance in mathematical relationships.
This idea is crucial for simplifying expressions like \( \frac{1}{a^{-3}} \). In this expression, \( a^{-3} \) becomes \( \frac{1}{a^3} \) because of the negative exponent rule. Now, you have \( \frac{1}{\frac{1}{a^3}} \), which is a complex fraction. By understanding the reciprocal rule, you know that dividing by \( \frac{1}{a^3} \) is the same as multiplying by \( a^3 \), helping us simplify the expression to just \( a^3 \). This transformation is pivotal as it turns convoluted algebra into straightforward arithmetic steps, aiding in a deeper appreciation of the elegance in mathematical relationships.
Other exercises in this chapter
Problem 67
In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ 12 a b \div 2 a b^{2} $$
View solution Problem 68
In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(50 a b^{4}\right)^{\frac{1}{2}} $$
View solution Problem 69
In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(16 a^{5} b^{6}\right)^{\frac{1}{4}} $$
View solution Problem 69
In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{3}{x^{4}} $$
View solution