Question

Simplify the expression. Write your answer using only positive exponents. \(5 \cdot 5^4\)

Short answer

\(5^5\)

Step-by-step solution

Identify the base and the exponents

In the expression \(5 \cdot 5^4\), the base is 5 and the exponents are implicitly 1 for the first 5 (since any number without an exponent is to the power of 1) and 4 for the second 5.

Apply the rule of exponents

Using the rule \(a^n \cdot a^m = a^{n+m}\), the expression can be simplified to \(5^{1+4}\).

Simplify

Simplify \(5^{1+4}\) to \(5^5\).

Key concepts

Rules of Exponents

When working with exponents, it's important to understand the various rules that govern how they are manipulated. One of the foundational rules is the Product of Powers rule, which states that when multiplying two expressions with the same base, you add the exponents. For example, when you have an expression such as
\(a^n \times a^m = a^{n+m}\)
Other rules include the Power of a Power rule, where \((a^n)^m = a^{n \times m}\), and the Quotient of Powers rule, where \(\frac{a^n}{a^m} = a^{n-m}\) when the base 'a' is the same and 'n' and 'm' are positive integers. Mastering these rules helps significantly in simplifying expressions with exponents.

Positive Exponents

Positive exponents indicate the number of times a base is used as a factor in a product. For instance, \(2^3 = 2 \times 2 \times 2 = 8\). In contrast, negative exponents represent the reciprocal of the base raised to the absolute value of the exponent (\(2^{-3} = 1/(2^3) = 1/8\)). When simplifying expressions, we aim to express them using only positive exponents. By applying the rules of exponents properly, negative exponents can usually be rewritten as positive ones, ensuring the expression is in its simplest form.

Exponent Simplification

Simplifying an expression with exponents can involve several steps, including combining like terms, converting negative exponents to positive, and applying the power rules. For instance, starting with the example \(5 \times 5^4\), the bases are the same, so you can add the exponents. Keep in mind that \(5 = 5^1\) due to the implicit exponent of 1.
Following the rule mentioned earlier:
\(5^1 \times 5^4 = 5^{1+4}\). This simplifies to \(5^5\). Remember to combine like terms wisely and watch out for potential opportunities to simplify further with roots or factorization.

Base and Exponent Identification

Identifying the base and exponent in an expression is a critical first step in simplifying it. The base is the number that is being multiplied by itself, and the exponent tells you how many times this operation occurs. In the expression \(5 \times 5^4\), '5' is the base. The first '5' is actually \(5^1\) since any number to the power of 1 is itself. Recognizing these components correctly enables the application of exponent rules and leads to successful simplification of the expression.