Lesson 20: Exponents, Rectangular Area, Part 1, Square Root

New Concept

An exponent shows repeated multiplication. It is used to calculate area, where finding a square's side length requires taking the square root.

What’s next

This card is just the foundation. Next, you'll tackle worked examples on simplifying exponents, finding area, and calculating square roots.

Property

An exponent shows how many times the base is to be used as a factor. In the expression $5^4$, the base is 5 and the exponent is 4.

Examples

• $5^3 = 5 \cdot 5 \cdot 5 = 125$
• $10^4 = 10 \cdot 10 \cdot 10 \cdot 10 = 10,000$
• $(\frac{1}{3})^2 = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}$

Explanation

Think of an exponent as a tiny instruction manual for a number! The base is the number you're working with, and the exponent tells you exactly how many times to multiply that base by itself. It’s a super handy shortcut for writing out long, repetitive multiplications, saving you time and making your math look sleek and professional.

Property

To find the area of a square or rectangle, use these formulas:

Examples

A rectangle with length 6 in. and width 4 in. has an area of $A = 6 \text{ in.} \cdot 4 \text{ in.} = 24 \text{ in.}^2$.
A square with a side length of 5 miles has an area of $A = (5 \text{ mi})^2 = 25 \text{ mi}^2$.
A square with a perimeter of 20 cm has sides of $20 \div 4 = 5$ cm, so its area is $A = (5 \text{ cm})^2 = 25 \text{ cm}^2$.

Explanation

Finding a shape's area is like calculating how many one-foot square tiles would cover its entire surface. Instead of counting each tile individually, you can use a simple trick: multiply the length by the width. For a perfect square where all sides are equal, you just multiply the side length by itself. It's the ultimate shortcut for measuring flat spaces!

Property

If we know the area of a square, we can find the length of its side by finding the square root of the area. The square root symbol is $\sqrt{\phantom{x}}$. For example, since a 3-by-3 square has an area of 9, the square root of 9 is 3.

Examples

To find the square root of 49, ask what number times itself equals 49. Since $7 \cdot 7 = 49$, we know that $\sqrt{49} = 7$.
Squaring a number and finding its square root are inverse operations, so they undo each other. For example, $\sqrt{10^2} = \sqrt{100} = 10$.
What is $\sqrt{144}$? We need a number that gives 144 when multiplied by itself. That number is 12, so $\sqrt{144} = 12$.

Explanation

Think of finding a square root as a math mystery! You are given the final area of a square, and your mission is to uncover the original side length. You need to ask yourself, "What number, when I multiply it by itself, gives me this area?" It’s the ultimate reverse-move of squaring a number, turning an area back into a length.