Problem 73 - Elementary and Intermediate Algebra

Question

Evaluate. $$ 45 \div 3^{2} x(x-1), \text { for } x=3 $$

Step-by-Step Solution

Answer

The value is 30.

Step 1: Understand the Expression

The given expression is \( 45 \div 3^{2} x (x - 1) \). We need to evaluate it for \( x = 3 \).

Step 2: Substitute the Value of x

Substitute \( x = 3 \) into the expression: \( 45 \div 3^{2} \times 3 \times (3 - 1) \).

Step 3: Simplify the Exponents

Evaluate \( 3^{2} \) which is \( 3 \times 3 = 9 \). Now the expression is \( 45 \div 9 \times 3 \times 2 \).

Step 4: Perform the Division

First, divide 45 by 9 to get \( 45 \div 9 = 5 \). So the expression simplifies to \( 5 \times 3 \times 2 \).

Step 5: Multiply the Remaining Terms

Now multiply the remaining terms: \( 5 \times 3 = 15 \) and \( 15 \times 2 = 30 \).

Step 6: Write the Final Answer

The final value of the expression for \( x = 3 \) is 30.

Key Concepts

substituting values in expressions

In algebra, substituting values means replacing a variable with a specific number. In the exercise, we had the expression \( 45 \div 3^{2} x (x - 1) \) and needed to evaluate it for \( x = 3 \). This step involves:
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• Identifying the variable(s) in the expression.
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• Replacing each occurrence of the variable with the given number.
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For our example, substituting \( x = 3 \) transforms the expression into \( 45 \div 3^{2} \times 3 \times (3 - 1) \). It's essential to correctly substitute the values to proceed with the following simplification steps.

order of operations

Order of operations in mathematics is crucial for correctly simplifying expressions. The standard order can be remembered using the acronym PEMDAS:
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• Parentheses
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• Exponents
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• Multiplication and Division (from left to right)
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• Addition and Subtraction (from left to right)
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Applying the order of operations to our expression after substitution \( 45 \div 3^{2} \times 3 \times (3 - 1) \):
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• First, calculate the exponent \( 3^{2} \) which is 9, now we have \( 45 \div 9 \times 3 \times 2 \).
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• Next, perform the division: \( 45 \div 9 = 5 \).
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• Finally, perform the multiplications in sequence: \( 5 \times 3 = 15 \) and \( 15 \times 2 = 30 \).
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Following these steps ensures our final answer is accurate.

simplifying expressions

Simplifying expressions involves performing arithmetic operations in the correct order to reduce an expression to its simplest form. After substituting \( x = 3 \) into the expression \( 45 \div 3^{2} \times 3 \times (3 - 1) \):
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• First, evaluate the parentheses \( (3 - 1) = 2 \).
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• Then simplify the exponent \( 3^{2} = 9 \).
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• Now we have \( 45 \div 9 \times 3 \times 2 \).
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• Next perform the division: \( 45 \div 9 = 5 \).
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• Multiply the remaining terms: \( 5 \times 3 = 15 \) and \( 15 \times 2 = 30 \).
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This process shows the transformation from a complex expression to a simple numerical value.

exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. It is written as \( a^{b} \), where \( a \) is the base and \( b \) is the exponent. In our exercise, we had \( 3^{2} \):
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• The base is 3.
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• The exponent is 2.
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• Calculating \( 3^{2} \) means multiplying 3 by itself: \( 3 \times 3 = 9 \).
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Understanding exponentiation helps simplify expressions by dealing with powers before other operations, allowing us to follow the order of operations properly.