Problem 54 - Elementary Algebra

Question

Use the order of operations to simplify the quantities for the following problems. $$ 3^{4}+2^{4}(1+5)^{3} $$

Step-by-Step Solution

Answer

Answer: The simplified result of the expression is 3537.

Step 1: Evaluate the expressions inside the parentheses

The expression that needs to be simplified is: $$ 3^{4} + 2^{4}(1 + 5)^{3} $$ First, we need to evaluate the expressions inside the parentheses: $$ (1+5) $$ Add 1 and 5 together: $$ 6 $$ Now, our expression looks like this: $$ 3^{4} + 2^{4} \cdot 6^{3} $$

Step 2: Evaluate the exponents

We have three exponents in our expression: $$ 3^4,\ 2^4,\ \text{and}\ 6^3 $$ Evaluate each exponent: $$ 3^4 = 81 \\ 2^4 = 16 \\ 6^3 = 216 $$ Now, substitute the evaluated exponents back into the expression: $$ 81+16\cdot216 $$

Step 3: Perform multiplication

We now need to multiply 16 and 216 together: $$ 16\cdot 216 = 3456 $$ Our expression now looks like this: $$ 81+3456 $$

Step 4: Perform addition

Finally, add 81 to 3456: $$ 81+3456=3537 $$ So, the simplified result of the given expression is: $$ 3^{4}+2^{4}(1+5)^{3}=3537 $$

Key Concepts

Working with Exponents

Exponents come into play when a number needs to be multiplied by itself a certain number of times. It's a way to express repeated multiplication.
For example, when you see \(3^4\), it means \(3\times3\times3\times3\).
Understanding how to deal with exponents is crucial in simplifying expressions.Here are a few key facts about exponents:

• An exponent shows how many times a number, known as the base, is multiplied by itself.
• A simple notation like \(x^n\) represents \(x\) raised to the power \(n\).
• Simplifying exponents is often the first step after dealing with parentheses, as they are one of the most powerful components in an expression.

In our exercise, simplifying \(3^4\) and \(2^4\) is necessary before moving forward since it reduces the complexity of the next operations.

Understanding Parentheses

Parentheses are used to denote which operations in an expression should be performed first. They tell us exactly what to focus on within the given mathematical problem. In this way, they sidestep any confusion and ensure clarity.For our expression \((1+5)\), the parentheses make it clear that addition should be performed before any exponentiation or multiplication. Here are a few insights about parentheses:

• Computations within parentheses take precedence over other operations. Always resolve these first.
• If an expression contains nested parentheses, start with the innermost set.
• Certainly, parentheses act as a grouping tool to clarify which numbers and operations go together.

Once we compute \((1+5)\) to get \(6\), we move on to evaluate the remaining parts of the expression.

Performing Multiplication

After dealing with exponents and parentheses, multiplication is often the next step in the order of operations, commonly known as BODMAS/BIDMAS (Brackets/Order of Divisions, Multiplications, Additions, Subtractions). In our simplified expression, the step involves multiplying the results of the exponents: \(16\) and \(216\). Following the order:

• We compute \(16 \cdot 216\) and achieve \(3456\).
• This operation is straightforward but essential to maintaining accuracy.
• Multiplication comes before addition, so performing it correctly at this point is key to the problem.

Keep track of computations to ensure errors don't cascade down the steps.

Finalizing with Addition

After all other operations, addition is performed last according to the order of operations. This step brings together the results from previous simplifications into the final value.In our ongoing example, we have arrived at \(81 + 3456\). This addition gives us the final numerical value:

• Simply add the intermediate step numbers together to achieve this.
• Accuracy matters; take your time to ensure you sum accurate values.
• Addition is the last straightforward step, serving as a capstone to the complex process.

With the final result being \(3537\), you conclude simplifying the expression using correct order of operations.