Question

Find each square root. Round to the nearest tenth, if necessary. \(\sqrt{64}\)

Short answer

The square root of 64 is 8.

Step-by-step solution

Identify the Expression

The problem is asking for the square root of 64, denoted as \(\sqrt{64}\). A square root of a number \(x\) is a number \(y\) such that \(y^2 = x\).

Calculate the Square Root

Recognize that 64 is a perfect square. Calculate the square root by recognizing a number which, when squared, results in 64. Check that \(8 \times 8 = 64\). Hence, \(\sqrt{64} = 8\).

Round to the Nearest Tenth

Since 8 is already a whole number and not a decimal, there's no need to round it to the nearest tenth. The square root is exact and does not require further rounding.

Key concepts

Perfect Square

A perfect square is a number that can be expressed as the product of an integer multiplied by itself. In simpler terms, it's what you get when you square a whole number. For example:

• 4 is a perfect square since it's the result of multiplying 2 by 2 (2 × 2).
• 9 is also a perfect square because 3 squared (3 × 3) equals 9.
• 64, being the number in our problem, is a perfect square because \[8 \times 8 = 64\]

Understanding perfect squares is important when finding square roots because if a number is a perfect square, its square root is always a whole number. Recognizing perfect squares helps speed up solving these types of problems.

Calculating Square Roots

Calculating the square root involves identifying a number that, when multiplied by itself, gives the original number. It's essentially the inverse of squaring a number. Let's consider the calculation of \(\sqrt{64}\):

• The square root of 64 is the number which, when squared, equals 64.
• This number is 8, since \[8 \times 8 = 64\]

Recognizing perfect squares speeds up finding square roots. For numbers that aren’t perfect squares, finding the square root might require estimation, use of a calculator, or numerical methods.

Exact vs. Non-exact Roots

When your square root calculation results in a whole number, it's an exact root, like with 64. If it involves decimals, finding the square root might require rounding or approximation.

Rounding Numbers

Rounding numbers is a mathematical technique used to simplify numbers. The process involves making a number easier to work with by keeping its value close to the original. When you round a number, you often change it to one with fewer decimal places.

Rounding to the Nearest Tenth

• This involves looking at the digit in the hundredths place to decide whether to round the tenth upwards or keep it the same.
• For example, if you have 5.67, you look at 7 (the hundredths). Since it's 5 or more, you round up, making it 5.7.

In our example with 64, the square root is \(8\), a whole number without any decimals, so no rounding was necessary.However, being comfortable with rounding is useful for more complex problems, especially when dealing with non-perfect squares, where the roots are not exact whole numbers.