Question

If \(a^{2}=49\) and \(b^{2}=64,\) which of the following CANNOT be a value of \(a+b ?\) F. \(-15\) G. \(-1\) H. \(\quad 1\) J. 15 K. 113

Short answer

A. -15 B. -1 C. 1 D. 15 E. 113 Answer: E. 113

Step-by-step solution

Find the possible values of a and b based on their squares

Using the information provided, we have \(a^2=49\) and \(b^2=64\). Since these are perfect squares, we can determine their square roots as follows: $a = \pm\sqrt{49} \\ a = \pm7$ $b = \pm\sqrt{64} \\ b = \pm8$ So, the possible values of \(a\) are 7, -7, and the possible values of \(b\) are 8, -8.

Find all possible values of a + b

Based on the possible values of \(a\) and \(b\) we found in Step 1, we can now find all possible values of \(a+b\). There are four possibilities for \(a+b\): 1. \(a+b = 7+8 = 15\) 2. \(a+b = 7-8 = -1\) 3. \(a+b = -7+8 = 1\) 4. \(a+b = -7-8 = -15\)

Determine which value cannot be the value of a + b

Now that we have all possible values of \(a+b\), we can compare them with the given options. F. \(-15\) (possible) G. \(-1\) (possible) H. \(\quad 1\) (possible) J. 15 (possible) The option that is not in the list is: K. 113 (not possible) Therefore, the correct answer is \(\boxed{\text{K. } 113}\), which cannot be a value of \(a+b\).

Key concepts

Perfect Squares

When dealing with perfect squares in mathematics, we refer to numbers that are the product of a whole number multiplied by itself. For example, when you multiply 7 by 7, the result is 49, which is a perfect square. Similarly, 64 is a perfect square because it is 8 times 8.

Understanding perfect squares is critical when solving equations where a variable is squared, such as in the expression \(a^2=49\). Recognizing that 49 is the square of 7 allows us to determine that \(a\) could be either 7 or -7, because both \(7^2\) and \((-7)^2\) equal 49. This is a foundational concept in not only algebra but also in functions and geometry where square values are common.

Square Roots

The square root operation is essentially the opposite of squaring a number. If you have a perfect square like 49 or 64, the square root will give you the original number that was squared. The square root of 49 is 7, because 7 times 7 equals 49, and the square root of 64 is 8.

In algebra, solving for a variable often involves taking the square root of both sides of an equation. When we write \(\sqrt{49}\), we're asking, 'What number times itself gives 49?' It's important to remember that there are always two possible answers: a positive and a negative one. Thus, \(\sqrt{49}\) can be either +7 or -7, because squaring either of these values will return 49.

Algebraic Operations

Algebraic operations include addition, subtraction, multiplication, and division applied to algebraic expressions and equations. When you're tasked with finding the value of \(a+b\) and you know the values of \(a^2\) and \(b^2\), you need to apply algebraic operations systematically. First, identify the potential values of \(a\) and \(b\) by taking square roots of their squares. Then, thoroughly explore the combinations of \(a+b\): adding and subtracting these potential values.

Here's a practical tip: always check all possible combinations of positive and negative values to ensure you haven't missed any potential solutions. This thorough approach not only fortifies your understanding of algebraic operations but also prepares you for more complex equations you'll encounter in advanced mathematics.