Problem 138 - College Algebra

Question

Fill in each box to make the statement true. $$\sqrt{x}=5 x^{7}$$

Step-by-Step Solution

Answer

The value of x that makes the equation true is \(x = 0\).

Step 1: Making sense of the equation

The equation given is \( \sqrt{x}=5x^{7} \). This equation implies that the square root of a certain number 'x' is equal to 5 times the 7th power of 'x'.

Step 2: Understanding the condition for the solution

Now, to find a legitimate 'x', we must ensure that the expression for 'x' must be non-negative. Hence, it is important to understand that this equation would only be valid for positive values, including zero, of 'x'.

Step 3: Finding the value(s) of x

Given that the square root of x equals 5 times the 7th power of x, this equation can only be true for \(x = 0\). That's because the square root of 0 is 0, while 5 times any power of 0 is also 0. However, any non-zero number raised to the 7th power would be much larger than its square root, so only 0 makes the original equation true.

Key Concepts

Square Root

The square root of a number is a value that, when multiplied by itself, gives the original number. In simpler terms, if you have a number like 9, the square root is the value that, when squared (multiplied by itself), equals 9. Therefore, the square root of 9 is 3, because \(3 \times 3 = 9\).

In equations, the square root is represented by the radical symbol (\(\sqrt{\;}\)). For example, \(\sqrt{x}\) represents the square root of \(x\).

It's crucial to note that square roots are generally considered in terms of non-negative numbers. This is because the product of two positive numbers, or two negative numbers, gives a positive result. Thus, when dealing with real numbers, the square root always returns the principal or positive square root.

Power Functions

Power functions are expressions where a number, called the base, is raised to a certain power or exponent. For instance, \(x^2\) is a power function where \(x\) is the base and 2 is the exponent. This means multiplying \(x\) by itself once (\(x \times x\)).

Power functions can vary greatly:

• When the exponent is positive, the value of the power function generally grows as the base increases.
• When the exponent is negative, the value decreases as the base increases, because you are dealing with fractions or reciprocals.

In our given equation, \(5x^7\), the base \(x\) is raised to the power of 7, and then multiplied by 5. This means, \(x\) is multiplied by itself six more times, and then the entire expression is multiplied by 5. Such functions can lead to rapid increases in value as \(x\) becomes larger.

Non-Negative Solutions

Non-negative solutions refer to solutions that are zero or any positive number. When dealing with certain functions, like the square root, having non-negative solutions is vital.

This is because the square root of a negative number results in an imaginary number, which most real-life problems do not deal with.

• For \(\sqrt{x} = 5x^7\), if \(x\) is negative, \(\sqrt{x}\) does not yield a real number.
• If \(x\) is zero or positive, the square root of \(x\) remains well-defined.

In the context of our equation, non-negative solutions ensure that both the left and the right side of the equation are valid and real, allowing us to determine \(x = 0\) as the only viable solution.