Problem 35 - Prealgebra

Question

Simplify each expression as much as possible. $$10 \div\left(\frac{1}{2}\right)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 40.

Step 1: Identify the Expression to Simplify

The expression given is \( 10 \div \left(\frac{1}{2}\right)^{2} \). We need to simplify this expression.

Step 2: Simplify the Parentheses

First, simplify the expression inside the parentheses. \( \left(\frac{1}{2}\right)^{2} \) means \( \frac{1}{2} \times \frac{1}{2} \), which equals \( \frac{1}{4} \).

Step 3: Simplify the Division

The expression is now \( 10 \div \frac{1}{4} \). Dividing by a fraction is the same as multiplying by its reciprocal. So, \( 10 \div \frac{1}{4} = 10 \times 4 \).

Step 4: Perform the Multiplication

Now, multiply: \( 10 \times 4 = 40 \).

Key Concepts

Simplifying Expressions

Simplifying an expression involves reducing it to its simplest form. It makes the expression easier to understand and work with. When simplifying, you follow the rules of arithmetic and algebra to break down or combine like terms.
Simplification usually involves eliminating parentheses, applying exponent rules, and combining like terms. In the example exercise, the expression is simplified by:

• Beginning with applying the exponent inside the parentheses to compute \(\left(\frac{1}{2}\right)^2\), which gives \(\frac{1}{4}\) by multiplying the fraction \(\frac{1}{2}\) by itself.
• After obtaining \frac{1}{4}\, the next simplification step makes the expression easier by changing the division of the fraction into a multiplication, which is often more straightforward to manage.

Simplification is a vital skill in math as it allows you to solve problems more easily and ensures that your answers are in their cleanest form.

Division by Fractions

Division by fractions can be tricky at first, but it's based on a simple principle: dividing by a fraction is the same as multiplying by its reciprocal. This conversion makes calculations straightforward.
To find the reciprocal of a fraction, you swap its numerator and denominator. For instance, the reciprocal of \(\frac{1}{4}\) is \(4\), because \(\frac{1}{4}\) is flipped to become \(4\) (or \(\frac{4}{1}\)).
To solve \(10 \div \frac{1}{4}\), convert the division into multiplication using the reciprocal:

• Change the division sign to multiplication: \(10 \times 4\).
• This transformation often simplifies the arithmetic, turning a complex problem into a simpler multiplication question.

Understanding how to divide by fractions is essential in prealgebra as it builds the foundation for more advanced math concepts.

Multiplication

Multiplication is one of math's foundational operations, crucial for simplifying expressions, especially after converting division by fractions into multiplication.
It involves repeated addition; for example, \(10 \times 4\) is like adding 10 four times in a row, which equals 40. In the context of division by fractions, you have already swapped division for multiplication using the reciprocal.
For instance:

• To complete \(10 \div \frac{1}{4}\), rewrite and solve it as \(10 \times 4\).
• Simply calculate as you would in any multiplication problem to find the solution.

Familiarity with multiplication not only helps in straightforward calculations but also makes tackling complex problems easier as you progress into higher math.