Problem 59 - Algebra 1: Concepts and Skills

Question

Write the expression as a single power of the base. $$ (-5) \cdot(-5)^{8} $$

Step-by-Step Solution

Answer

The simplified expression as a single power of the base is \((-5)^9\).

Step 1: Identify the base and exponents

The base in this problem is -5 and it is raised to the powers of 1 and 8. So, we can rewrite the expression as \((-5)^1\) and \((-5)^8\).

Step 2: Apply the rule of exponents for multiplication

The rule of exponents for multiplication states that when multiplying two exponents of the same base, the exponents can be added. Put mathematically, \(a^m \cdot a^n = a^{m+n}\). In our case, the base is -5, so applying this rule, we get \((-5)^1 \cdot (-5)^8 = (-5)^{1 + 8}\).

Step 3: Simplify the Expression

We can now simplify the expression. \((-5)^{1 + 8} = (-5)^9\).

Key Concepts

Understanding Exponents

Exponents represent how many times a number, known as the base, is multiplied by itself. For instance, in the expression \((-5)^8\), the base is \(-5\) and the exponent is 8. This means that \(-5\) is multiplied by itself 8 times. Exponents provide a convenient way to express repeated multiplication. By understanding how exponents work, you can simplify complex expressions and solve various mathematical problems easily.

Multiplication of Exponents

When dealing with multiplication, if you have exponents with the same base, you can use the rule of exponents for multiplication. The rule states that you can add the exponents together when multiplying like bases. This is expressed as \(a^m \cdot a^n = a^{m+n}\).

In our exercise, for the expression \((-5)^1 \cdot (-5)^8\), notice that the base is the same, which is \(-5\). Thus, you apply this rule to add up the exponents: \(1 + 8 = 9\). This transforms the expression into a single power: \((-5)^9\). By applying this rule, calculations become easier and more efficient.

Simplification of Expressions

Simplification involves reducing expressions to their simplest form. In our problem, after summing the exponents, the expression becomes \((-5)^9\), simplifying \((-5)^1 \cdot (-5)^8\) into a single power of the base \(-5\). This reduction helps to clarify the problem and makes it more manageable.

Simplification not only aids in computations but also in understanding mathematical relationships and connections. By getting familiar with simplification techniques, you can tackle more complex problems with confidence and ease.