Question

Simplify each expression. \(\sqrt{(-3)^{2}}\)

Short answer

\(\backslash sqrt{(-3)^{2}} = 3\).

Step-by-step solution

Understand the Expression

The expression given is \(\backslash sqrt{(-3)^{2}}\). This involves taking the square root of the square of \-3\.

Square the Number

Calculate \((-3)^{2}\). This means multiplying -3 by itself: \((-3) \times (-3) = 9\).

Take the Square Root

Now, find the square root of 9. The square root of 9 is 3 because \(3 \times 3 = 9\).

Simplify the Expression

Thus, \(\backslash sqrt{(-3)^{2}} = \backslash sqrt{9} = 3\).

Key concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). In mathematical notation, the square root is represented using the radical sign \(\sqrt{}\). Square roots are always non-negative. So \(\sqrt{9} = 3\) and not \(\sqrt{9} = -3\), even though \( (-3) \times (-3) = 9 \) as well.

If you're working with negative numbers under the square root, remember that the square root of any positive number squared will always be positive. In other words, \(\sqrt{(-3)^2}\) simplifies to \(\sqrt{9}\), which equals 3.

Exponents

Exponents show how many times a number, known as the base, is multiplied by itself. In our example, \( (-3)^2 \), -3 is the base, and 2 is the exponent. This means we multiply -3 by itself: \( (-3) \times (-3) = 9 \).

It's important to note the effect of the exponent on negative bases. A negative number raised to an even exponent (like 2, 4, 6, etc.) will always result in a positive value. This happens because the pairings of the negative signs cancel each other out. For example:

• \( (-2)^2 = 4 \)
• \( (-2)^4 = 16 \)

On the other hand, raising a negative number to an odd exponent (like 1, 3, 5, etc.) will result in a negative value. For example:

• \( (-2)^3 = -8 \)
• \( (-2)^5 = -32 \)

Multiplication

Multiplication is one of the basic arithmetic operations. It involves combining equal groups. For example, \(3 \times 4 \) means 3 groups of 4. When dealing with negative numbers, remember these simple rules:

• The product of two positive numbers is positive: \(3 \times 4 = 12\)
• The product of two negative numbers is positive: \( (-3) \times (-4) = 12 \)
• The product of a positive number and a negative number is negative: \(3 \times -4 = -12\)

In the given exercise, \( (-3) \times (-3) \) becomes positive because the two negative signs cancel each other out.

Algebraic Simplification

Algebraic simplification is the process of reducing expressions to their simplest form. This often involves combining like terms and using arithmetic operations.

Let's take the expression \( \sqrt{(-3)^2} \) that we simplified. Here's how we break it down step-by-step:

• Simplify the exponent: \( (-3)^2 = 9 \)
• Take the square root of the result: \( \sqrt{9} = 3 \)

This shows the original expression simplifies to a single number, 3.

Algebraic simplification can involve other operations and properties as well, such as the distributive property or combining like terms. Always follow the order of operations and handle each part of the expression step-by-step.