Problem 25 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \frac{5^{6}}{5^{3}} $$

Step-by-Step Solution

Answer

The simplified quotient is 125.

Step 1: Identify the base and the exponents

Here, the base is 5 and the exponents are 6 and 3. We have \(5^{6}\) divided by \(5^{3}\).

Step 2: Apply the rule of exponents for division

For division, we subtract the exponent of the denominator from the exponent of the numerator, when they have the same base. So it becomes \(5^{(6-3)}\).

Step 3: Simplify the exponent

We subtract the exponents. So, \(5^{(6-3)}\) becomes \(5^{3}\).

Step 4: Calculate the value

\(5^{3}\) simplifies to 125.

Key Concepts

Rules of Exponents

Exponents can be intimidating at first, but they follow specific rules that make working with them much easier. Rules of exponents are essentially shortcuts for performing operations with exponential expressions.

For example, when you multiply two exponents with the same base, you add the exponents. In mathematical terms, if you have \( a^m \times a^n \) you get \( a^{m+n} \). Conversely, when dividing exponents with the same base as shown in the exercise, you subtract the exponents: \( a^m \div a^n = a^{m-n} \).

There are more rules for different scenarios as well:

• A number raised to the power of zero is always 1, so \( a^0 = 1 \).
• To raise a power to another power, you multiply the exponents, so \( (a^m)^n = a^{m \cdot n} \).
• When multiplying exponents with different bases, you cannot add the exponents unless you simplify the bases to be the same.
• Distributing an exponent over a multiplication within parentheses means raising each element inside to the exponent, like \( (ab)^n = a^n b^n \).

Understanding these rules simplifies many operations involving exponents and makes solving exponential expressions much more approachable.

Exponential Expressions

An exponential expression is a mathematical notation that implies the repeated multiplication of a number - known as the base - by itself. The exponent represents the number of times the base is used as a factor.

For instance, \( 5^6 \) tells us that the number 5 is multiplied by itself 6 times. This is far more concise than writing 5 \times 5 \times 5 \times 5 \times 5 \times 5. Exponents are not just shorthand; they are a powerful way to express very large or very small numbers - such as in scientific notation where exponential expressions are essential.

Grasping exponential expressions involve understanding some properties:

• Positive exponents indicate standard multiplication.
• Negative exponents signify division (i.e., \( a^{-n} = \frac{1}{a^n} \) ).
• A fractional exponent like \( a^{1/n} \) corresponds to the nth root of a.

Operations with exponents, therefore, can be transformed into much simpler terms when we apply the appropriate rules of exponents.

Division of Powers

The concept of division of powers comes into play when two exponential expressions with the same base are divided. As mentioned in the rules of exponents, the key operation here is to subtract the exponents.

If you have an expression like \( \frac{a^m}{a^n} \), where both the numerator and denominator share the same base \(a\), then you can reduce this by keeping the base and subtracting the exponent in the denominator from the exponent in the numerator, resulting in \(a^{m-n}\). This simplification can drastically reduce the complexity of the expression and is an essential technique in algebra.

In the case of our textbook example, \( \frac{5^6}{5^3} \), both terms have the base 5. We subtract the exponent in the denominator (3) from the exponent in the numerator (6), which simplifies to \( 5^{6-3} = 5^3 \). Finally, solving \( 5^3 \) we find that it equals 125. Breaking down expressions with the \( a^m \div a^n \) format becomes much more manageable with this straightforward approach.