Problem 4 - College Algebra and Calculus: An Applied Approach

Question

Evaluate the expression. Write fractional answers in simplest form.\(\frac{5^{7}}{5^{5}}\)

Step-by-Step Solution

Answer

The expression \(\frac{5^{7}}{5^{5}}\) simplifies to 25.

Step 1: Identify the expression

Look at the given expression, here it is \(\frac{5^{7}}{5^{5}}\). We have two numbers with the same base (5) divided by one another. The exponents are 7 and 5 respectively.

Step 2: Apply the rule of exponents

When dividing terms with the same base, we subtract the exponents. Here, 5 is the common base, so subtract the exponents as follows: \(5^{7-5}\).

Step 3: Solve for the exponent

Next, subtract the exponents 7 and 5 which equals 2, making the expression \(5^2\).

Step 4: Evaluate the simplified expression

Finally, we calculate \(5^2\) which is \(5*5 = 25\).

Key Concepts

Exponent rules

Exponents are a way to express repeated multiplication. When we have an expression like \( 5^7 \), the number 5 is the base and 7 is the exponent. It means we multiply 5 by itself 7 times: \( 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \).

To make calculations easier, there are several important exponent rules to remember:

• **Multiplication Rule**: When you multiply terms with the same base, you add their exponents: \( a^m \times a^n = a^{m+n} \).
• **Division Rule**: When you divide terms with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• **Power of a Power Rule**: When you raise an exponent to another power, you multiply the exponents: \( (a^m)^n = a^{m\times n} \).
• **Zero Exponent Rule**: Any non-zero base raised to the power of zero is 1: \( a^0 = 1 \)}

Understanding these rules helps in simplifying expressions efficiently, such as the one in our exercise where dividing \( 5^7 \) by \( 5^5 \) simplifies to \( 5^{7-5} = 5^2 \).

Keeping these rules in your toolkit will make working with exponents much easier.

Simplifying expressions

Simplifying expressions involves using rules to break down complex terms into more manageable forms.

In the exercise, we simplified the expression \( \frac{5^{7}}{5^{5}} \) using the division rule for exponents. By subtracting the exponents with the same base, we found \( 5^{7-5} \), which equals \( 5^2 \).

Simplification often requires identifying common bases, like we did with 5, and applying the correct rule. It can also involve:

• **Combining like terms** where coefficients of similar expression parts are added together.
• **Factoring** which is rewriting expressions as a product of its factors.
• **Canceling out terms** to further reduce fractions by finding common factors in numerators and denominators.

The key is to make expressions simpler and easier to work with without changing their value. Keeping practice in simplifying helps solve not just mathematical, but also algebraic expressions efficiently.

Fractional forms

Fractions represent parts of a whole. They often consist of a numerator (top part) and a denominator (bottom part).

Fractions can also involve variables and exponents, making simplification essential.

When simplifying fractional forms:

• Ensure that the fraction is in its simplest form by dividing both numerator and denominator by their greatest common divisor (GCD).
• Sometimes changing the fraction to decimal form helps visualize comparisons or approximations, especially in practical scenarios.

The idea is to express the fraction in the most reduced form without altering its value. In our exercise, \( \frac{5^{7}}{5^{5}} \) simplifies to a whole number rather than a fraction through rules of exponents, showing that not every fractional form ends up as a fraction.

By getting comfortable with fractional forms, one can solve a variety of problems concerning parts of quantities, making such mathematical tasks more intuitive.