Problem 43 - Prealgebra

Question

Simplify each expression as much as possible. $$24 \div\left(\frac{2}{5}\right)^{2}+25 \div\left(\frac{5}{6}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression equals 186.

Step 1: Evaluate the Exponents

First, let's evaluate each exponentiation separately. The expression \( \left(\frac{2}{5}\right)^{2} \) equals \( \frac{2}{5} \times \frac{2}{5} = \frac{4}{25} \). Similarly, \( \left(\frac{5}{6}\right)^{2} \) equals \( \frac{5}{6} \times \frac{5}{6} = \frac{25}{36} \).

Step 2: Perform the Division

Now we need to divide 24 by the result from Step 1, \( \frac{4}{25} \). Dividing by a fraction is the same as multiplying by its reciprocal, so \( 24 \div \frac{4}{25} = 24 \times \frac{25}{4} = 6 \times 25 = 150 \). Do the same for the second expression: 25 divided by \( \frac{25}{36} \) gives \( 25 \times \frac{36}{25} = 36 \), because \( \frac{25}{25} = 1 \).

Step 3: Add the Results

Finally, add the results from Step 2 together: 150 from the first part and 36 from the second part. The sum is \( 150 + 36 = 186 \).

Key Concepts

Exponents

Exponents are a way to represent repeated multiplication of a number by itself. They give us a compact method to write terms that multiply the same base number many times. For instance, the expression \((\frac{2}{5})^2\) means that the fraction \(\frac{2}{5}\) is multiplied by itself: \(\frac{2}{5} \times \frac{2}{5}\).

• When dealing with exponents, always identify the base and the power. The base is the number or expression that is being multiplied, and the exponent, or power, tells us how many times the base is used as a factor.
• For fractions, apply the exponent to both the numerator and the denominator separately. This means \((\frac{2}{5})^2\) becomes \(\frac{2^2}{5^2}\) which simplifies to \(\frac{4}{25}\).
• Exponents can also appear with negative or fractional powers, which convey division or root operations respectively, but for now, we focus on positive integer exponents.

Understanding exponents helps simplify expressions more effectively, taking care of the powers before moving to other arithmetic operations.

Dividing Fractions

Dividing fractions is an important concept in arithmetic and algebra that often confuses students. The key to dividing fractions is understanding how to multiply by the reciprocal.

• To divide by a fraction \(\frac{a}{b}\), you multiply by its reciprocal \(\frac{b}{a}\). This changes the division into a multiplication problem, which is easier to handle.
• For example, when the exercise asks us to perform \(24 \div \frac{4}{25}\), it's equivalent to \(24 \times \frac{25}{4}\). As shown: \(24 \times \frac{25}{4} = 6 \times 25 = 150\).
• This method works because multiplying by a reciprocal effectively "cancels out" the original fraction, leaving a simplification of the original number.

Practicing turning division into multiplication through reciprocals helps streamline many fraction problems in mathematics.

Order of Operations

In mathematics, the order of operations is crucial to correctly solving problems that involve multiple steps. Often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication/Division (from left to right), Addition/Subtraction (from left to right), this set of rules ensures consistent results.

• Always start with operations inside Parentheses. In our context, this means simplifying any expressions contained within brackets or parentheses first. This includes handling any powers or roots.
• Follow parentheses by handling Exponents, as seen in the first step of evaluating \((\frac{2}{5})^2\) and \((\frac{5}{6})^2\).
• Next, tackle Multiplication and Division. These operations are evaluated from left to right. In the given exercise, this involved dividing 24 by \((\frac{2}{5})^2\) and 25 by \((\frac{5}{6})^2\) after calculating the power results.
• Finally, proceed with Addition and Subtraction. In the exercise, the final step was to sum up the two results from the divisions to get the final answer of 186.

Being aware of and applying the proper order of operations safeguards against careless mistakes and leads to accurate simplification of complex expressions.