Problem 46
Question
Use the order of operations to simplify each expression. $$-3(-6+8)^{3}-5(-3+5)^{3}$$
Step-by-Step Solution
Verified Answer
The simplified expression is \(-64\).
1Step 1: Simplify Inside Parentheses
Before we can proceed with the order of operations, we must first simplify the expression within parentheses. So, you simplify \(-6+8\) and \(-3+5\) which equals \(2\) and \(2\) respectively.
2Step 2: Apply the Power Rule
Next step is to take care of the power rule, the 'I' / 'O' in BIDMAS / BODMAS. Therefore, calculate the cube of both \(2\)’s. This gives \(2^{3} = 8\) and \(2^{3} = 8\).
3Step 3: Simplify Equation
The next step is to simplify the equation by multiplying \(8\) to \(-3\) and \(-5\) respectively. \(-3*8 = -24\) and \(-5*8= -40\).
4Step 4: Add the Results
The final step is to add \(-24\) to \(-40\). Therefore, \(-24 + -40 = -64\).
Key Concepts
Simplify ExpressionsPower RuleBIDMAS/BODMAS
Simplify Expressions
When you come across a complex mathematical expression, your goal is to break it down into a simpler form. Simplifying expressions is a fundamental skill in algebra that allows us to reduce an equation to its most basic components, making it easier to understand and solve.
Simplification may involve combining like terms, eliminating parentheses, or reducing fractions. For example, in the problem \( -3(-6+8)^3 - 5(-3+5)^3 \), you start by addressing the arithmetic inside the parentheses before moving onto other operations. This makes the subsequent steps more manageable and reduces the risk of error.
By simplifying the components within the parentheses \( -6+8 \) to \( 2 \) and \( -3+5 \) to \( 2 \), the problem becomes easier to solve, as you are then dealing with smaller numbers or simpler expressions.
Simplification may involve combining like terms, eliminating parentheses, or reducing fractions. For example, in the problem \( -3(-6+8)^3 - 5(-3+5)^3 \), you start by addressing the arithmetic inside the parentheses before moving onto other operations. This makes the subsequent steps more manageable and reduces the risk of error.
By simplifying the components within the parentheses \( -6+8 \) to \( 2 \) and \( -3+5 \) to \( 2 \), the problem becomes easier to solve, as you are then dealing with smaller numbers or simpler expressions.
Power Rule
A key concept when dealing with exponential expressions is the power rule. This principle is crucial for simplifying expressions that involve powers or exponents.
The power rule states that to raise a number to a power, you multiply the number by itself as many times as the exponent indicates. For instance, in the expression \( 2^3 \), you would multiply \( 2 \times 2 \times 2 \) to get \( 8 \). In our exercise, both instances of \( 2 \) are raised to the third power, \( 2^3 \), and simplifying these according to the power rule gives us a value of \( 8 \) for each.
It's essential to accurately apply the power rule before combining the terms in an algebraic expression, as it directly affects the outcome of the solution.
The power rule states that to raise a number to a power, you multiply the number by itself as many times as the exponent indicates. For instance, in the expression \( 2^3 \), you would multiply \( 2 \times 2 \times 2 \) to get \( 8 \). In our exercise, both instances of \( 2 \) are raised to the third power, \( 2^3 \), and simplifying these according to the power rule gives us a value of \( 8 \) for each.
It's essential to accurately apply the power rule before combining the terms in an algebraic expression, as it directly affects the outcome of the solution.
BIDMAS/BODMAS
BIDMAS/BODMAS is an acronym used to remember the order of operations in mathematics: Brackets, Indices (powers and roots), Division and Multiplication, and Addition and Subtraction. Understanding BIDMAS/BODMAS is critical for correctly simplifying expressions.
The exercise \( -3(-6+8)^3 - 5(-3+5)^3 \) requires us to apply BIDMAS/BODMAS as follows:
The exercise \( -3(-6+8)^3 - 5(-3+5)^3 \) requires us to apply BIDMAS/BODMAS as follows:
- First, deal with what's inside the Brackets: Simplify \( -6+8 \) and \( -3+5 \) to \( 2 \) each.
- Next, apply the rule of Indices: Calculate the cube (third power) of \( 2 \) to get \( 8 \) for each expression.
- Then, address the Multiplication by -3 and -5 respectively.
- Finally, carry out the Addition (or in this case, adding negative numbers) to combine the results, which completes the simplification process to arrive at the solution \( -64 \) for the given problem.
Other exercises in this chapter
Problem 45
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