Problem 15
Question
Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. $$ \left(\frac{x^{2} y^{4} z^{7}}{a^{5} b}\right)^{9} $$
Step-by-Step Solution
Verified Answer
Question: Simplify the expression \(\left(\frac{x^{2} y^{4} z^{7}}{a^{5} b}\right)^{9}\).
Answer: \(\frac{x^{18}y^{36}z^{63}}{a^{45}b^{9}}\)
1Step 1: Apply the power rule for quotients
Apply the power rule for quotients to raise the entire expression to the 9th power.
$$
\left(\frac{x^{2} y^{4} z^{7}}{a^{5} b}\right)^{9} = \frac{(x^{2} y^{4} z^{7})^{9}}{(a^{5} b)^{9}}
$$
2Step 2: Apply the power rule for products to the numerator
Apply the power rule for products to raise each term in the numerator to the 9th power.
$$
(x^{2} y^{4} z^{7})^{9} = (x^{2})^{9}(y^{4})^{9}(z^{7})^{9}
$$
3Step 3: Apply the power rule for products to the denominator
Apply the power rule for products to raise each term in the denominator to the 9th power.
$$
(a^{5} b)^{9} = (a^{5})^{9}(b)^{9}
$$
4Step 4: Simplify the powers in the numerator
Simplify the powers in the numerator by multiplying the exponents.
$$
(x^{2})^{9}(y^{4})^{9}(z^{7})^{9} = x^{18}y^{36}z^{63}
$$
5Step 5: Simplify the powers in the denominator
Simplify the powers in the denominator by multiplying the exponents.
$$
(a^{5})^{9}(b)^{9} = a^{45}b^{9}
$$
6Step 6: Write the simplified expression
Combine the simplified numerator and denominator to write the final simplified expression.
$$
\frac{x^{18}y^{36}z^{63}}{a^{45}b^{9}}
$$
Key Concepts
Simplifying Algebraic ExpressionsExponent RulesAlgebraic Quotient
Simplifying Algebraic Expressions
Simplifying algebraic expressions is a fundamental skill in algebra. The goal is to make complex expressions easier to interpret and work with by reducing them to their simplest form. This process often includes combining like terms, factoring, expanding expressions, and using exponent rules. Like terms are those that have the same variables raised to the same power. For instance, in the expression \(2x^3 + 3x^3\), the two terms can be combined to \(5x^3\) because they both contain the variable \(x\) raised to the third power.
When dealing with algebraic quotients, such as \(\frac{x^2}{x^5}\), we subtract exponents if the base is the same, simplifying to \(x^{-3}\) or \(\frac{1}{x^3}\). This is because division is considered the inverse of multiplication in terms of exponent rules. Simplification can often lead to expressions that are easier to evaluate, differentiate, or integrate, which is essential in calculus and other areas of advanced mathematics.
When dealing with algebraic quotients, such as \(\frac{x^2}{x^5}\), we subtract exponents if the base is the same, simplifying to \(x^{-3}\) or \(\frac{1}{x^3}\). This is because division is considered the inverse of multiplication in terms of exponent rules. Simplification can often lead to expressions that are easier to evaluate, differentiate, or integrate, which is essential in calculus and other areas of advanced mathematics.
Exponent Rules
Exponent rules, sometimes called 'laws of exponents', are a set of rules that govern how to simplify expressions involving powers of numbers or variables. The power rule, for example, states that when you raise an expression to a power, you multiply the exponents. In mathematical terms, this is written as \( (x^m)^n = x^{m\cdot n} \). This rule is fundamental when simplifying expressions such as \( (x^2)^9 = x^{2\cdot9} = x^{18} \).
Other important exponent rules include the product rule \( x^m \cdot x^n = x^{m+n} \), and the quotient rule \( \frac{x^m}{x^n} = x^{m-n} \), which students used in our original exercise to simplify the algebraic quotient. Understanding and applying these rules correctly are critical steps in manipulating algebraic expressions efficiently. Exponents also play a vital role in more advanced mathematics, including polynomial equations, exponential growth and decay models, and in expressing derivatives and integrals in calculus.
Other important exponent rules include the product rule \( x^m \cdot x^n = x^{m+n} \), and the quotient rule \( \frac{x^m}{x^n} = x^{m-n} \), which students used in our original exercise to simplify the algebraic quotient. Understanding and applying these rules correctly are critical steps in manipulating algebraic expressions efficiently. Exponents also play a vital role in more advanced mathematics, including polynomial equations, exponential growth and decay models, and in expressing derivatives and integrals in calculus.
Algebraic Quotient
An algebraic quotient refers to the division of two algebraic expressions or polynomials. It takes the form of a fraction with a numerator and denominator that contains variables and possibly exponents. Simplifying an algebraic quotient often involves applying exponent rules and simplifying expressions within the numerator and the denominator before dividing them.
In the context of our exercise, students simplified the expression \( \left(\frac{x^{2} y^{4} z^{7}}{a^{5} b}\right)^{9} \) using the power rule for quotients, which is the process of distributing the exponent to both the numerator and denominator to simplify the entire expression. The result, \( \frac{x^{18}y^{36}z^{63}}{a^{45}b^{9}} \), is significantly simpler and can be further manipulated or evaluated if necessary. Learning to work with algebraic quotients is important for solving equations and understanding how variables relate to one another when they are divided.
In the context of our exercise, students simplified the expression \( \left(\frac{x^{2} y^{4} z^{7}}{a^{5} b}\right)^{9} \) using the power rule for quotients, which is the process of distributing the exponent to both the numerator and denominator to simplify the entire expression. The result, \( \frac{x^{18}y^{36}z^{63}}{a^{45}b^{9}} \), is significantly simpler and can be further manipulated or evaluated if necessary. Learning to work with algebraic quotients is important for solving equations and understanding how variables relate to one another when they are divided.
Other exercises in this chapter
Problem 15
For the following problems, write the appropriate relation symbol \((=,)\) in place of the \(*\). $$ 9[4+3(8)] * 6[1+8(5)] $$
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Find each quotient $$ \frac{a^{7}}{a} $$
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For the following problems, write each of the quantities using exponential notation. \(x\) to the eighth
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Use the distributive property to rewrite each of the following quantities. $$3(2+1)$$
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