Question

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

Short answer

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

Step-by-step solution

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: $$ (a^{5} b^{5}) = (ab)^{5}. $$

Simplify the rewritten expression

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

Key concepts

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.