Problem 51 Simplify. See Examples 1 through... [FREE SOLUTION]

Chapter 2: Problem 51

Simplify. See Examples 1 through $10$. $$ \left(10-4^{2}\right)^{2} $$

Short Answer

Expert verified

The simplified form is 36.

Step by step solution

Evaluate the Exponent

Firstly, evaluate the exponent inside the parentheses. The expression inside the parentheses is $ 4^2 $. Since $ 4^2 = 16 $, replace $ 4^2 $ with 16.

Simplify Inside the Parentheses

Now, substitute the value found in Step 1 back into the expression: $ 10 - 16 $. Calculate the result of this subtraction to simplify the expression inside the parentheses: $ 10 - 16 = -6 $.

Apply the Square

Take the result from Step 2 and apply the square (as indicated by the outer exponent in the original problem). Since the expression is now $ (-6)^2 $, calculate the square: $(-6)^2 = 36$.

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Key Concepts

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

Evaluating Exponents

When working with exponents, it's essential to understand what they represent. Exponents tell us how many times to multiply a number by itself. For instance, when you see $4^2$, this means $4 \times 4$, which equals 16. This is called "squaring" the number because we multiply it by itself. Evaluating exponents is the first crucial step before simplifying any expression that contains them.

Remember:

Exponents appear as the tiny number above the base number (e.g., in $4^2$, 4 is the base, and 2 is the exponent).
Evaluate exponents before any addition or subtraction within the parentheses.
This helps simplify expressions into smaller, more manageable numbers.

Parentheses Simplification

Once the exponent is evaluated, the next step is to focus on simplifying inside the parentheses. Parentheses indicate the part of an expression that must be solved first and works as a grouping tool in math. For the expression $10 - 16$, you should subtract 16 from 10.

This subtraction is straightforward but crucial to obtaining the correct simplified form of what is inside the parentheses.

Always resolve operations inside parentheses first, as it dictates the correct order of mathematical operations known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
Make sure to keep working in the correct order without skipping steps.

Recognizing parentheses' role in simplifying complex equations will ease even the most challenging problems.

Applying Exponents

After simplifying inside the parentheses, the final step is applying the exponent given outside the parentheses. This involves taking the simplified value and raising it to the power indicated. In this example, once we have $-6$ from the subtraction in the parentheses, the next step is to square it because of the exponent 2, now outside the parentheses.

Be careful to apply the exponent to the resulting value, respecting any signs. In mathematical terms, $(-6)^2$ is indeed $-6 \times -6$, which results in 36. Note that squaring a negative number yields a positive result.

This is because multiplying two negative numbers results in a positive number.
Reapplying the exponents correctly ensures the solution accurately represents the given expression.

Mastering how to apply exponents helps in simplifying various algebraic expressions efficiently.

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91影视

Simplify. See Examples 1 through \(10\). $$ \left(10-4^{2}\right)^{2} $$

Short Answer

Step by step solution

Evaluate the Exponent

Simplify Inside the Parentheses

Apply the Square

Key Concepts

Evaluating Exponents

Parentheses Simplification

Applying Exponents

One App. One Place for Learning.

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