Problem 72
Question
In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{3 a^{0} b^{-3}}{a^{-1} b^{-3}} $$
Step-by-Step Solution
Verified Answer
The quotient simplifies to \(3a\).
1Step 1: Apply the Zero and Negative Exponent Rule
Begin by simplifying the expression using the zero and negative exponent rules. Recall that any number raised to the power of 0 is 1, and the negative exponent rule states that \(x^{-n} = \frac{1}{x^n}\). Thus:\[3a^0b^{-3} = 3 \cdot 1 \cdot \frac{1}{b^3} = \frac{3}{b^3}\]and for the denominator:\[a^{-1}b^{-3} = \frac{1}{a} \cdot \frac{1}{b^3} = \frac{1}{ab^3}\].
2Step 2: Simplify the Fraction by Dividing
Next, divide the expression in the numerator by the expression in the denominator. Division of fractions is the same as multiplying by the reciprocal:\[\frac{3/b^3}{1/(ab^3)} = 3b^3 \times \frac{ab^3}{1} = 3ab^6\].
3Step 3: Final Simplification
After multiplying across, the fraction simplifies to a single product with no denominator:\[3a \].
Key Concepts
Zero Exponent RuleNegative Exponent RuleSimplification of Expressions
Zero Exponent Rule
In mathematics, the zero exponent rule simplifies expressions where the exponent is zero. This rule states that any number or variable, except zero itself, raised to the power of zero is 1. For example, \( a^0 = 1 \). This looking simple, mighty rule, simplifies expressions by converting terms with zero exponents into '1', which often helps reduce the complexity of expressions.
So whenever you spot a term like \( a^0 \) in your expressions, replace it with 1. Remember, this does not apply to zero because \( 0^0 \) is considered undefined.
Here's how this plays out in our exercise: the term \( a^0 \) in the numerator simplifies to 1, thus directly affecting the outcomes in multi-variable expressions, making them easier to handle.
So whenever you spot a term like \( a^0 \) in your expressions, replace it with 1. Remember, this does not apply to zero because \( 0^0 \) is considered undefined.
Here's how this plays out in our exercise: the term \( a^0 \) in the numerator simplifies to 1, thus directly affecting the outcomes in multi-variable expressions, making them easier to handle.
Negative Exponent Rule
The negative exponent rule is another powerful tool in simplifying mathematical expressions. This rule explains how to handle terms with negative exponents. You transform these by taking the reciprocal of the base and then changing the sign of the exponent to positive. Mathematically, this is stated as \( x^{-n} = \frac{1}{x^n} \).
Applying this rule helps convert a difficult expression into an easier one to manage. For instance, \( b^{-3} \) becomes \( \frac{1}{b^3} \). This inversion aids in removing negativity from the exponents, making simplification steps easier to manage.
In our problem, the term \( b^{-3} \) both in the numerator and denominator becomes fractional. This essential adjustment allows us to eliminate negative exponents and transform expressions efficiently.
Applying this rule helps convert a difficult expression into an easier one to manage. For instance, \( b^{-3} \) becomes \( \frac{1}{b^3} \). This inversion aids in removing negativity from the exponents, making simplification steps easier to manage.
In our problem, the term \( b^{-3} \) both in the numerator and denominator becomes fractional. This essential adjustment allows us to eliminate negative exponents and transform expressions efficiently.
Simplification of Expressions
Simplification is the key to solving complex mathematical problems, making them more manageable and understandable. When you simplify expressions, you reduce them to their simplest form using mathematical rules and operations. This often involves applying exponent rules, combining like terms, and performing arithmetic operations.
In the context of our example, simplification begins by first transforming exponents using the zero and negative exponent rules. Next, the focus shifts to simplifying the fraction by dividing the numerator by the denominator. This step involves multiplying by the reciprocal when fractions are present.
After transforming the entire expression, the terms \( 3b^3 \) in the numerator get multiplied by the reciprocal of the denominator, which is \( ab^3 \). The expression simplifies further to \( 3ab^6 \), demonstrating a final simplification step with no fraction present.
In the context of our example, simplification begins by first transforming exponents using the zero and negative exponent rules. Next, the focus shifts to simplifying the fraction by dividing the numerator by the denominator. This step involves multiplying by the reciprocal when fractions are present.
After transforming the entire expression, the terms \( 3b^3 \) in the numerator get multiplied by the reciprocal of the denominator, which is \( ab^3 \). The expression simplifies further to \( 3ab^6 \), demonstrating a final simplification step with no fraction present.
- Replace terms with zero exponents by '1'.
- Transform terms with negative exponents using reciprocals.
- Multiply and combine like terms to achieve a single expression without a denominator.
Other exercises in this chapter
Problem 71
In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{36}{9 x^{-5}} $$
View solution Problem 72
In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \left(\frac{-32 x^{10}}{y^{4}}\right)^{\frac{1}
View solution Problem 73
In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \frac{8^{\frac{1}{4}} a^{\frac{5}{6}} b^{\frac{
View solution Problem 73
In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{20 x^{0} y^{-5}}{4 x^{-1} y^{5}} $$
View solution