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Chapter 6: Problem 34

Write with a single exponent. $$ a^{5} b^{5} $$

Short Answer

Expert verified
Question: Express the given expression with a single exponent: $a^{5} b^{5}$. Answer: $(ab)^{5}$.

Step by step solution

01

Rewrite the expression using the exponent rule

According to the exponent rule: \((ab)^x = a^x \cdot b^x,\) we can rewrite the given expression as: $$ (a^{5} b^{5}) = (ab)^{5}. $$
02

Simplify the rewritten expression

The simplified expression with a single exponent is: $$ (ab)^{5}. $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Rules
Exponent rules help us manage and simplify expressions with powers. When you see an expression like \(a^5 b^5\), it can be rewritten using the power of a product rule. This rule states that \((ab)^x = a^x \cdot b^x\). Essentially, when two terms have the same exponent, you can combine them under one exponent.
This approach is incredibly useful:
  • It reduces complexity by combining terms.
  • Makes calculations easier, especially with multiple variables.
The exponent rules are foundational in algebra and help simplify many types of problems.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operators. They can involve addition, subtraction, multiplication, division, and exponentiation. In the expression \(a^5 b^5\), \(a\) and \(b\) are variables, and 5 is the exponent applied to both.
Expressions allow us to represent mathematical situations:
  • Variables stand in for unknown values or can represent general numbers.
  • Exponents show repeated multiplication of a variable.
Understanding each part of an expression is key to manipulating and simplifying it using various algebraic techniques.
Simplifying Expressions
Simplifying expressions is about making them easier to work with, often by reducing the number of terms or using the fewest operations possible. In simpler terms, it's about cleaning up math problems to see them at their most basic form.
To simplify \(a^5 b^5\) using the exponent rules, we combined the variables under a single power, resulting in \((ab)^5\).
Benefits of simplifying expressions:
  • Simpler forms are easier to evaluate or compute.
  • It makes solving equations more straightforward.
  • It reveals the underlying structure of the expression.
By combining terms and using exponent rules, simplifying becomes a straightforward and eye-opening process.

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Most popular questions from this chapter

Find a conjugate of each expression and the product of the expression with the conjugate. $$ 7 \sqrt{2}-2 \sqrt{7} $$

Write each expression without parentheses. Assume all variables are positive. $$ (c f)^{9} $$

Without a calculator, decide whether the quantities are positive or negative. $$ -31^{66} $$

Write each expression without parentheses. Assume all variables are positive. $$ \left(\frac{c^{3}}{d}\right)^{4} $$

Without a calculator, decide whether the quantities are positive or negative. $$ -4^{3} $$
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