Problem 138 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solution to the equation \(\sqrt{x}=5x^7\) is \(x = 1\).

Step 1: Remove the square root

To eliminate the square root on the left side, we square both sides of the equation: \((\sqrt{x})^2 = (5x^7)^2\). Simplifying this gives us \(x = 25x^{14}\)

Step 2: Isolate x

Next, isolate \(x\). We divide both sides by \(25x^{14}\) to find \(x\). The left side becomes \(x/25x^{14} = 1\) and the right side becomes \(25x^{14} / 25x^{14} = 1\). Simplifying, we find \(x = 1/25x^{13}\).

Step 3: Simplify

Lastly, simplify the equation by multiplying both sides by \(25x^{13}\). This gets rid of the denominator on the left and gives us \(x = 1\).

Key Concepts

Solving Equations

Solving equations is like unwrapping a present to find what's inside: the value of the unknown variable. Your goal is to isolate the variable on one side of the equation. This means getting *x* by itself so you can find its value.
To do this, you often need to perform a series of operations in a specific order. These might include

• adding or subtracting terms,
• multiplying or dividing by numbers or variables, and
• applying inverse operations like squaring and taking square roots.

Each step should simplify the equation further, helping you edge closer to the solution. It's like peeling back layers until you have the core, which is the answer to the problem.

Square Roots

The square root operation is the opposite of squaring a number. When you see \( \sqrt{x} \) this means you're looking for a number that multiplies by itself to give you \(x\).
For example, \( \sqrt{9} = 3 \) since \(3 \times 3 = 9\).
To remove a square root in an equation, like \( \sqrt{x} = 5x^7 \), simply square both sides of the equation. By doing this, you eliminate the square root and help reveal the hidden variable.
For instance, \( \left(\sqrt{x}\right)^2 = \left(5x^7\right)^2 \) simplifies to \(x = 25x^{14} \), making it easier to solve for *x*. Remember, squaring both sides helps unlock the next value in your solution!

Exponents

Exponents represent repeated multiplication. When you see \( x^n \), it means you're multiplying \(x\) by itself \(n\) times. Exponents make it easier to write long multiplication chains in a compact form.
Consider solving equations involving exponents by leveraging properties like:

• Multiplying exponents where \( x^a \times x^b = x^{a+b} \)
• Dividing exponents where \( x^a / x^b = x^{a-b} \)
• Raising a power to another power \( \left(x^a\right)^b = x^{ab} \)

In our example, \( (5x^7)^2 \) becomes \(25x^{14} \) since \(5^2 = 25\) and \((x^7)^2 = x^{14}\).
Using these properties, you can simplify parts of equations, making it less complicated to manage.

Simplifying Expressions

Simplifying expressions means making them easier to understand and solve. It's about transforming a complex problem into a simpler one.
You simplify by:

• Combining like terms (e.g., \(2x + 3x = 5x\)
• Using mathematical properties, such as the distributive property
• Reducing fractions by finding common factors

In the last step of our example, simplifying \( \frac{x}{25x^{14}} \) resulted in \( \frac{1}{25x^{13}} \), making it easier to solve for \(x\).
Simplifying helps you see the path to the solution more clearly. By consistently simplifying as you solve, you clarify the problem and enhance your understanding of the mathematical landscape.