Problem 9 - Algebra 1: Concepts and Skills

Question

Use the power of a power property to write the expression as a single power of the base. \(\left(y^{4}\right)^{5}\)

Step-by-Step Solution

Answer

The simplified form of the expression \(\left(y^{4}\right)^{5}\) is \(y^{20}\)

Step 1: Identify the base and the powers

In the expression \(\left(y^{4}\right)^{5}\), the base 'y' is raised to the power 4, and then this is raised to the power 5. So, the base is 'y', the first power is 4, and the second power is 5.

Step 2: Apply the power of a power property

According to the power of a power property, when an exponent is raised to another exponent, the exponents are multiplied. The formula for this is \((b^m)^n = b^{m*n}\) So therefore, the expression becomes \(y^{4*5}\)

Step 3: Simplify the expression

Now, simply perform the multiplication in the exponent. This results in \(y^{20}\)

Key Concepts

Exponents

Exponents are a key part of mathematics that help us express repeated multiplication succinctly. When you see an expression like \(y^4\), it means that the base \(y\) is being multiplied by itself four times: \(y \times y \times y \times y\). Exponents make it easier to write and work with such expressions.
Moreover, exponents follow specific rules or properties that allow us to manipulate them in algebra confidently.

• The product of powers property: \(b^m \times b^n = b^{m+n}\)
• The power of a power property: \((b^m)^n = b^{m*n}\)
• The power of a product property: \((ab)^m = a^m \times b^m\)

Understanding these properties helps in simplifying and solving complex algebraic expressions easier and faster.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like plus and minus signs). They represent mathematical problems in a concise form. In an expression like \(5y + 3\), \(y\) is a variable which can be any number.
Variables add flexibility, allowing you to create a general formula that can apply to multiple situations. This is particularly useful in real-world applications where exact numbers aren't known.
Exponents can be part of these algebraic expressions, as seen in \(\left(y^{4}\right)^{5}\). Here, the variable \(y\) is raised to a power, adding another layer of complexity and utility.
Algebra is essential because it forms the foundation of higher-level math and is used in fields such as engineering, computer science, and economics. When dealing with algebraic expressions, remember to follow the order of operations (PEMDAS/BODMAS) to simplify correctly.

Simplifying Expressions

Simplifying expressions is about making them as straightforward as possible without changing their value. This involves combining like terms, reducing fractions, and applying mathematical properties or rules, such as those for exponents.
For the expression \(\left(y^{4}\right)^{5}\), simplification requires the use of the power of a power property of exponents.

• According to this property, you multiply the exponents: \( (y^4)^5 = y^{4 \times 5} \).
• Then, compute the multiplication: \( 4 \times 5 = 20 \).
• Hence, the expression simplifies to \( y^{20} \).

Simplifying makes expressions easier to read and work with, which is especially useful when solving algebraic equations or performing calculations in science and engineering. By understanding and applying the relevant properties, you can efficiently reduce complex expressions to simpler forms.