Question

Write each expression as a power raised to a power. There may be more than one correct answer. $$ 5^{2 y} $$

Short answer

Based on the provided solution, rewrite the expression \(5^{2 y}\) as a power raised to another power. Answer: \((5^2)^y\)

Step-by-step solution

Rewrite the exponent as a product

First, we will rewrite the exponent of the given expression as a product which includes \(y\). We can do this by rewriting \(2y\) as \(2 \cdot y\): $$ 5^{2 y} = 5^{2 \cdot y} $$

Apply the exponent rule of powers

Now we will apply the exponent rule of powers, which states that \((a^m)^n = a^{m \cdot n}\). In this case, \(a = 5\), \(m = 2\), and \(n = y\). Using this rule, we can rewrite the expression as follows: $$ 5^{2 \cdot y} = (5^2)^y $$

Final expression

After applying the exponent rule of powers, we have the following expression: $$ (5^2)^y $$ This is the final expression for the given problem, as a power raised to a power.

Key concepts

Exponent Rules

When dealing with exponents, it's important to understand the various rules that govern their operations. These rules simplify expressions, making it easier to solve mathematical problems. Here are the main exponent rules:

• **Product of Powers Rule**: When multiplying two powers with the same base, you simply add the exponents. For example, \(a^m \cdot a^n = a^{m+n}\).
• **Quotient of Powers Rule**: When dividing two powers with the same base, subtract the exponents. This looks like \(\frac{a^m}{a^n} = a^{m-n}\), assuming \(a eq 0\).
• **Zero Exponent Rule**: Any nonzero number raised to the power of zero equals one. For instance, \(a^0 = 1\).
• **Negative Exponent Rule**: A negative exponent signifies taking the reciprocal of the base. So, \(a^{-n} = \frac{1}{a^n}\).

Understanding these rules is essential for working with complex expressions and will help you to see how values interact with each other in algebraic contexts.
Mastering them can make handling mathematical expressions a breeze.

Power of a Power Rule

The "Power of a Power Rule" is a handy tool when you are working with expressions involving multiple layers of exponents. This rule comes into play when you have an exponent raised to another exponent, and it states:

• \((a^m)^n = a^{m \cdot n}\)

It's like multiplying the two exponents together.
For example, if you had \((3^2)^4\), you would multiply 2 by 4 to get \(3^8\).
This rule is particularly useful when simplifying expressions and can save time when calculating large powers.
In the original exercise, this rule was used to rewrite the expression \(5^{2y}\) as \((5^2)^y\). By recognizing this structure, we simplified the problem into a more manageable form.

Mathematical Expressions

Mathematical expressions consist of numbers, variables, and operators that stand for a value. They are a universal language in math, representing complex ideas simply and effectively.
An expression can be as simple as a number or a variable, like 7 or \(x\), or can involve complicated combinations of these elements.
When dealing with expressions that include exponents, like \(5^{2y}\), it's important to understand how to manipulate them effectively.

• **Variables** - Symbols like \(x\) or \(y\) used to represent unspecified numbers or values.
• **Constants** - Fixed values like 5 or \(\pi\) that do not change.
• **Operators** - Include symbols like +, -, ×, ÷ that denote the operations to be performed.

By breaking down expressions into simpler parts and applying rules like the power of a power rule, you can solve them more easily.
This makes working with expressions efficient, focusing on simplification and clarity.