Problem 4
Question
Name both square roots of each of the following numbers. $$ 64 $$
Step-by-Step Solution
Verified Answer
Answer: The two square roots of 64 are 8 and -8.
1Step 1: Identify the positive square root
For the positive square root, we're looking for a positive number that, when multiplied by itself, results in 64.
Since \(8 \times 8 = 64\), the positive square root of 64 is 8.
2Step 2: Identify the negative square root
For the negative square root, we're looking for a negative number that, when multiplied by itself, results in 64.
Since \((-8) \times (-8) = 64\), the negative square root of 64 is -8.
Both square roots of the number 64 are \(\pm 8\).
Key Concepts
AlgebraPerfect SquaresRadicals
Algebra
Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It provides a systematic way for describing and analyzing relationships between variable quantities. One fundamental aspect of algebra is the use of operations, such as addition, subtraction, multiplication, and division, in conjunction with variables to solve equations.
In the case of square roots, algebra helps us to understand the relationship between a number and its square root. For example, if we have the equation \( x^2 = 64 \), algebraic operations can be used to solve for \( x \) by finding the square root of 64. This is where recognizing 'perfect squares' comes into play, as well as understanding 'radicals' to represent roots. It's crucial to understand that an equation involving square roots can have both a positive and a negative solution, as showcased in the provided exercise where both \( +8 \) and \( -8 \) were identified as square roots of 64.
In the case of square roots, algebra helps us to understand the relationship between a number and its square root. For example, if we have the equation \( x^2 = 64 \), algebraic operations can be used to solve for \( x \) by finding the square root of 64. This is where recognizing 'perfect squares' comes into play, as well as understanding 'radicals' to represent roots. It's crucial to understand that an equation involving square roots can have both a positive and a negative solution, as showcased in the provided exercise where both \( +8 \) and \( -8 \) were identified as square roots of 64.
Perfect Squares
Perfect squares are the product of a whole number multiplied by itself. The term 'perfect' signifies that they're exact squares and not fractions or decimals. Common examples of perfect squares include numbers like 1 (\(1^2\)), 4 (\(2^2\)), 9 (\(3^2\)), and so on.
Knowing your perfect squares is an essential skill in algebra because they often appear in equations and can simplify the problem-solving process. When a problem involves finding the square root of a perfect square, like 64, recognizing that 64 is \(8^2\) makes finding the solution straightforward. However, not all numbers are perfect squares, and sometimes we may need to estimate the square root or express it as an irrational number, represented by a radical when the number inside the square root cannot be simplified to a perfect square.
Knowing your perfect squares is an essential skill in algebra because they often appear in equations and can simplify the problem-solving process. When a problem involves finding the square root of a perfect square, like 64, recognizing that 64 is \(8^2\) makes finding the solution straightforward. However, not all numbers are perfect squares, and sometimes we may need to estimate the square root or express it as an irrational number, represented by a radical when the number inside the square root cannot be simplified to a perfect square.
Radicals
Radicals are mathematical symbols used to represent the root of a number. The most common radical is the square root, but radicals can also represent cube roots, fourth roots, and so on. The radical symbol \( \sqrt{\phantom{x}} \) specifically refers to square roots. When we use a radical without a number outside it, we are typically referring to the square root.
The use of radicals extends the concept of 'perfect squares' because not all square roots result in whole numbers. For example, while \( \sqrt{64} \) is 8, \( \sqrt{2} \) isn't a whole number and we leave it as \( \sqrt{2} \) unless we need a decimal approximation. Understanding how radicals work is crucial to simplifying expressions that contain roots and to operating with roots in algebraic equations. Notably, when we take the square root of a number, we must consider both the positive and negative roots, because \( a^2 = b \) means that \( a \) can be both the positive and negative square root of \( b \). In our exercise, the number 64 has both \( +8 \) and \( -8 \) as its square roots, usually expressed as \( \pm8 \) in radical form.
The use of radicals extends the concept of 'perfect squares' because not all square roots result in whole numbers. For example, while \( \sqrt{64} \) is 8, \( \sqrt{2} \) isn't a whole number and we leave it as \( \sqrt{2} \) unless we need a decimal approximation. Understanding how radicals work is crucial to simplifying expressions that contain roots and to operating with roots in algebraic equations. Notably, when we take the square root of a number, we must consider both the positive and negative roots, because \( a^2 = b \) means that \( a \) can be both the positive and negative square root of \( b \). In our exercise, the number 64 has both \( +8 \) and \( -8 \) as its square roots, usually expressed as \( \pm8 \) in radical form.
Other exercises in this chapter
Problem 4
For the following problems, simplify the expressions. $$ \sqrt{11} \sqrt{11} $$
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Simplify the square root expressions. $$ \frac{\sqrt{196(x+7)^{8}}}{\sqrt{2(x+7)^{3}}} $$
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Find each of the following products. $$ \sqrt{8 m^{5} n} \sqrt{20 m^{2} n} $$
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Simplify each square root. $$ \sqrt{144 x^{4} y^{80}(b+5)^{16}} $$
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