Problem 78
Question
Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ a^{x} b^{y} c^{5 z} $$
Step-by-Step Solution
Verified Answer
Answer: The simplified expression is \(a^{x} b^{y} c^{5 z}\).
1Step 1: Apply the relevant algebraic rules
Use properties of exponents, radicals, or algebraic identities to simplify the expression.
2Step 2: State the simplified result
The simplified expression is Answer: The simplified expression is \(a^{x} b^{y} c^{5 z}\)..
Key Concepts
Product Rule of ExponentsExponent SimplificationAlgebraic Expressions
Product Rule of Exponents
The Product Rule of Exponents is a fundamental principle in algebra that allows us to simplify expressions with exponents. When you multiply two algebraic expressions with the same base, you can add the exponents. In mathematical terms, if you have a term like \( a^m \) multiplied by \( a^n \), the product rule tells us that \( a^m \cdot a^n = a^{m+n} \).
Understanding this rule is incredibly handy, especially when dealing with more complex algebraic expressions. It simplifies the process and reduces potential errors from excessive multiplication. To effectively apply the product rule, remember that the bases must be the same, and only then can you perform the exponent addition.
For example, if you had the expression \( x^3 \cdot x^4 \), applying the product rule would give you \( x^{3+4} \), which simplifies to \( x^7 \). This rule is an essential building block that paves the way for greater algebraic manipulating and simplification.
Understanding this rule is incredibly handy, especially when dealing with more complex algebraic expressions. It simplifies the process and reduces potential errors from excessive multiplication. To effectively apply the product rule, remember that the bases must be the same, and only then can you perform the exponent addition.
For example, if you had the expression \( x^3 \cdot x^4 \), applying the product rule would give you \( x^{3+4} \), which simplifies to \( x^7 \). This rule is an essential building block that paves the way for greater algebraic manipulating and simplification.
Exponent Simplification
Exponent Simplification is the process of reducing expressions with exponents to their simplest form. This involves using rules like the product rule, the quotient rule, and the power of a power rule to manipulate the expressions. Simplifying exponents involves a few key steps: be aware of the base, correctly apply exponent rules, and simplify any coefficients.
For instance, when you're given an expression like \( a^x \cdot a^y \), the simplification would just be \( a^{x+y} \) by the product rule we discussed earlier. In cases where you encounter a quotient, such as \( \frac{a^m}{a^n} \), you would subtract the exponents (\( a^{m-n} \)) according to the quotient rule of exponents. Simplification may seem daunting at first, but with practice, you can strengthen your ability to work with these algebraic expressions confidently.
For instance, when you're given an expression like \( a^x \cdot a^y \), the simplification would just be \( a^{x+y} \) by the product rule we discussed earlier. In cases where you encounter a quotient, such as \( \frac{a^m}{a^n} \), you would subtract the exponents (\( a^{m-n} \)) according to the quotient rule of exponents. Simplification may seem daunting at first, but with practice, you can strengthen your ability to work with these algebraic expressions confidently.
Algebraic Expressions
Algebraic Expressions are combinations of constants, variables, and operators (like +, -, ×, ÷) forming a mathematical phrase. They can include terms with exponents, which represent repeated multiplication of a base. These expressions are like sentences in the language of algebra, conveying values that depend on the variables involved.
Crucially, understanding how to manipulate these expressions with exponent rules allows for simplification and solution of complex problems. Always keep in mind variables can represent any number, so algebraic expressions are versatile tools for generalizing arithmetic and solving equations.
Returning to our exercise, the expression \(a^{x} b^{y} c^{5z}\) showcases variables a, b, and c raised to exponents. These kinds of expressions are fundamental in algebra and help develop skills necessary for more advanced mathematics such as calculus or physics. By mastering the manipulation of algebraic expressions, you pave the way for understanding broader concepts in mathematics and science.
Crucially, understanding how to manipulate these expressions with exponent rules allows for simplification and solution of complex problems. Always keep in mind variables can represent any number, so algebraic expressions are versatile tools for generalizing arithmetic and solving equations.
Returning to our exercise, the expression \(a^{x} b^{y} c^{5z}\) showcases variables a, b, and c raised to exponents. These kinds of expressions are fundamental in algebra and help develop skills necessary for more advanced mathematics such as calculus or physics. By mastering the manipulation of algebraic expressions, you pave the way for understanding broader concepts in mathematics and science.
Other exercises in this chapter
Problem 77
Find the value of \(\frac{(5-3)^{2}+(5+4)^{3}+2}{4^{2}-2 \cdot 5-1}\)
View solution Problem 78
For the following problems, write the expressions using exponential notation. \((y+2)\) cubed.
View solution Problem 78
Assuming the bases are not zero, find the value of \(\left(4 a^{2} b^{3}\right)\left(5 a b^{4}\right)\).
View solution Problem 79
For the following problems, write the expressions using exponential notation. \((a+2 b)\) squared minus \((a+3 b)\) to the fourth.
View solution