Problem 20 - Algebra 1

Question

Evaluate the expression. Write fractions in simplest form. $$ \frac{8^{3}}{8^{1}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is 64

Step 1: Understand the Applicable Exponent Law

The problem involves properties of exponents. When dividing numbers with the same base, the exponents are subtracted. This follows the rule \(a^{m}/a^{n} = a^{m-n}\) where \(a\) is the common base, \(m\) is the exponent of the number in the numerator and \(n\) is the exponent of the number in the denominator.

Step 2: Apply the Exponent Law

Using the rule from Step 1, the expression simplifies to \(8^{3-1}\). This results in \(8^{2}\).

Step 3: Simplify the Expression

Next, simplify the expression \(8^{2}\). Taking 8 to the power of 2 gives us 64.

Key Concepts

Simplifying Expressions

The process of simplifying expressions is about making complex mathematical expressions easier to understand and work with. This often involves reducing the number of terms, combining like terms, or applying mathematical properties to condense the expression into a more manageable form.

Consider the expression \(\frac{8^{3}}{8^{1}}\). To simplify this, we recognize that both numbers share the same base, which is 8. By reducing this expression, you're not just making your work cleaner; you're also sharpening your mathematical comprehension. So, as you face an expression such as this one, remember simplification is your tool for clarity and efficiency in mathematics.

Exponent Laws

Understanding the exponent laws, also known as the laws of indices, is crucial when working with expressions involving powers. These laws dictate how to handle multiplication, division, and powers of numbers with exponents.

For instance, the law for dividing powers with the same base states that you subtract the exponents: \(a^{m}/a^{n} = a^{m-n}\). In the given exercise, applying this law allows us to subtract the exponents directly: \(8^{3}/8^{1} = 8^{3-1} = 8^{2}\). Familiarity with these rules can greatly simplify many algebraic problems and is an essential skill in algebra.

Numerical Fractions

Numerical fractions represent parts of a whole and they are written with a numerator and a denominator. When working with fractions in algebra, it's important to simplify them to their lowest terms to make them easier to understand and compare.

In the context of the exercise, even though the initial expression is presented as a fraction, the application of exponent laws transitions it into a whole number. This move from a fractional form to an integer is an illustration of simplifying a numerical fraction, even though it might not look like the typical fraction we are used to seeing. Remember, simplifying makes calculations less cumbersome and often reveals the quantities in a more relatable context.