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Chapter 6: Problem 4

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

Short Answer

Expert verified
The square root of 64 is 8.

Step by step solution

01

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 \(y^2 = x\).
02

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, \(\sqrt{64} = 8\).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 脳 2).
  • 9 is also a perfect square because 3 squared (3 脳 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鈥檛 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.

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