Problem 16 $$\begin{aligned} (\sqrt{-7})^... [FREE SOLUTION]

Chapter 11: Problem 16

$$\begin{aligned} (\sqrt{-7})^{2} &=\sqrt{(-7)^{2}} \\ &=\sqrt{49} \\ &=7 \end{aligned}$$

Short Answer

Expert verified

The simplified expression $(\sqrt{-7})^{2}$ actually equals $-7$, not 7 as given in the equation. The correct equation should involve the absolute value of the square root to result in a positive value: $\\ \begin{aligned} |\sqrt{-7}|^{2} &= |-7| \\ &= 7 \end{aligned}$

Step by step solution

Apply exponent rules to the expression

In the given expression $(\sqrt{-7})^{2}$, we need to first apply the exponent to the square root. Recall that the square root of a number is the same as raising the number to the exponent $\frac{1}{2}$. So, $(\sqrt{-7})^{2}$ is equivalent to $(-7)^{\frac{1}{2}\cdot2}$.

Simplify the exponent

Now, we need to simplify the exponent. We have $(-7)^{\frac{1}{2}\cdot2}$, which simplifies to $(-7)^{1}$, as $\frac{1}{2}\cdot2$ equals 1.

Evaluate the simplified expression

We have now simplified the original expression to $(-7)^{1}$, which simply equals $-7$.

Compare the result to the given equation

Our resulting expression is $-7$. However, the given equation suggests that the value should be 7. This difference in sign indicates an error in the original equation. A correct equation should involve the absolute value of the square root, since squaring the square root of a negative number would result in a positive number. The correct equation should be: $\\ \begin{aligned} |\sqrt{-7}|^{2} &= |(-7)^{\frac{1}{2}\cdot2}| \\ &= |-7| \\ &= 7 \end{aligned}$

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Key Concepts

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

Square Roots

Square roots help us find a number that, when multiplied by itself, gives the original number. For example, $\sqrt{49}$ equals 7, because 7 multiplied by 7 is 49. When dealing with square roots, especially with negative numbers, things get a little tricky. You can typically take square roots of positive numbers and zero.

Positive numbers have a real square root.
Negative numbers do not have a real square root under standard conditions because there is no real number which squared gives a negative number. Instead, we use complex numbers, involving 'i', the imaginary unit.

In the context of our exercise, the square root of a negative number is managed by considering the absolute value of the number under the root, which ultimately represents a real positive outcome when squared.

Exponent Rules

Exponents signify the number of times a base is multiplied by itself. In the case of square roots, we can think of them as powers, specifically a power of one half.When you see $b^{m}\cdot b^{n} = b^{m+n}$, this rule of multiplying same bases helps simplify expressions. Similarly, $a^{m/n}$ translates to taking the n-th root of a, raised to the m-th power.For example, in $\sqrt{-7} = (-7)^{1/2}$ and squaring it gives $\left((-7)^{1/2}\right)^{2} = (-7)^1 = -7,$ gently nudging us towards the importance of evaluating carefully, especially with negative numbers.

Absolute Value

Absolute value tells us how far a number is from zero on the number line, without caring about direction. The absolute value of both -7 and 7 is 7. This explains why negative numbers, when squared, turn into positive results; they become their positive equivalent as distance to zero:

|7| = 7
|-7| = 7

In the original exercise, having absolute values helps correct an error in finding the result. Taking absolute value like $\left|\sqrt{-7}\right|^2$ ensures the outcome of our expression as 7, rather than insisting on keeping the negative sign, which isn't needed here.

Negative Numbers

Negative numbers are those less than zero, found on the left side of the number line. When you multiply or square negative numbers, the results can be surprising.

Multiplying two negatives: Negative times negative equals positive. So, (-2) 脳 (-3) = 6.
Squaring a negative: (-3)^2 results in 9, since the double negative multiplies to positive.

Understanding how negative numbers behave with exponents and absolute values ensures clear solutions for our math problems. As in the exercise, when we reconsider $\sqrt{-7}$ with its absolute value, negative results transform to positives upon completion.

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91影视

$$\begin{aligned} (\sqrt{-7})^{2} &=\sqrt{(-7)^{2}} \\ &=\sqrt{49} \\ &=7 \end{aligned}$$

Short Answer

Step by step solution

Apply exponent rules to the expression

Simplify the exponent

Evaluate the simplified expression

Compare the result to the given equation

Key Concepts

Square Roots

Exponent Rules

Absolute Value

Negative Numbers

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