Problem 46
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. $$ (2 x y)(3 y)\left(4 x^{2} y^{5}\right) $$
Step-by-Step Solution
Verified Answer
Question: Simplify the expression \((2xy)(3y)(4x^2 y^5)\).
Answer: The simplified expression is \(24x^3y^7\).
1Step 1: Distribute the terms
Multiply each term in the parentheses:
$$
(2xy)(3y)(4x^2 y^5) = (2\cdot3\cdot4)(x\cdot x^2)(y\cdot y\cdot y^5)
$$
2Step 2: Apply the product rule for exponents
Add the exponents of the variables with the same base:
$$
(2\cdot3\cdot4)(x\cdot x^2)(y\cdot y\cdot y^5) = 24x^{1+2}y^{1+1+5}
$$
3Step 3: Simplify the expression
Add the exponents and write the simplified expression:
$$
24x^{1+2}y^{1+1+5} = 24x^3y^7
$$
The simplified expression is \(24x^3y^7\).
Key Concepts
Product Rule of ExponentsQuotient Rule of ExponentsExponent OperationsAlgebraic Expressions
Product Rule of Exponents
When simplifying algebraic expressions that involve multiplication of variables with exponents, the product rule of exponents can simplify this process significantly. This rule states that when multiplying two expressions with the same base, you simply add the exponents together. For instance, when we look at the expression \( x^m \times x^n \), we can use the product rule to combine these expressions into one: \( x^{m+n} \).
In our exercise, we used the product rule to combine \( x \cdot x^2 \) into \( x^{1+2} \), similarly, we combined \( y \cdot y \cdot y^5 \) into \( y^{1+1+5} \). It's important to remember that this rule only applies to terms with the same base; terms with different bases are multiplied normally.
In our exercise, we used the product rule to combine \( x \cdot x^2 \) into \( x^{1+2} \), similarly, we combined \( y \cdot y \cdot y^5 \) into \( y^{1+1+5} \). It's important to remember that this rule only applies to terms with the same base; terms with different bases are multiplied normally.
Quotient Rule of Exponents
Just as the product rule of exponents assists in simplifying multiplication of exponential terms, the quotient rule comes in handy when dividing such terms. This rule says that when you have an expression where you're dividing two exponents with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Formally, \( \frac{x^m}{x^n} = x^{m-n} \).
Although our example does not specifically involve division, it's essential to understand how these rules complement each other. If the problem required us to divide, say \( \frac{x^3}{x^2} \), using the quotient rule would simplify it to \( x^{3-2} = x \). Remember, like the product rule, this only applies to terms with the same base.
Although our example does not specifically involve division, it's essential to understand how these rules complement each other. If the problem required us to divide, say \( \frac{x^3}{x^2} \), using the quotient rule would simplify it to \( x^{3-2} = x \). Remember, like the product rule, this only applies to terms with the same base.
Exponent Operations
Exponent operations encompass a variety of rules, including the product and quotient rules mentioned earlier, as well as the power rule, zero exponent rule, and negative exponent rule. These operations allow us to manipulate exponential expressions in order to simplify or solve algebraic problems.
When faced with exponents, it's crucial to assess which operation applies to the situation at hand and perform it correctly. For instance, raising an exponent to another power (\( (x^m)^n \)) calls for the power rule, where you multiply the exponents for the same base, giving you \( x^{mn} \). Always start by checking the bases of your exponential terms; this determines which exponent operation is applicable.
When faced with exponents, it's crucial to assess which operation applies to the situation at hand and perform it correctly. For instance, raising an exponent to another power (\( (x^m)^n \)) calls for the power rule, where you multiply the exponents for the same base, giving you \( x^{mn} \). Always start by checking the bases of your exponential terms; this determines which exponent operation is applicable.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. The purpose is to represent a value or set of values succinctly. These expressions become tools for solving equations, modeling real-life situations, and analyzing relationships between different quantities.
When simplifying expressions as in our example, understanding the order in which to apply operations is vital. We start by dealing with terms inside parentheses, then move on to exponents, and finally carry out multiplication or division from left to right. By understanding and applying rules like the product and quotient rules of exponents, we ensure that algebraic expressions are simplified correctly and efficiently.
When simplifying expressions as in our example, understanding the order in which to apply operations is vital. We start by dealing with terms inside parentheses, then move on to exponents, and finally carry out multiplication or division from left to right. By understanding and applying rules like the product and quotient rules of exponents, we ensure that algebraic expressions are simplified correctly and efficiently.
Other exercises in this chapter
Problem 45
For the following problems, rewrite each phrase using algebraic notation. 3 times \((x+y)\)
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Is there a smallest two digit real number? If so, what is it?
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Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers. $$
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For the following problems, select a number (or numbers) to show that \((a+b)^{2}\) is not generally equal to \(a^{2}+b^{2}\).
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