Evaluate The Expression:$\[ -1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right) \\]Use The Keypad To Enter The Answer In The Box.$\[ -1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right) = \square \\]

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Understanding the Expression

The given expression is −1+83÷(12⋅312)-1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right). To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Breaking Down the Expression

Let's break down the expression into smaller parts to make it easier to evaluate:

  • 838 \sqrt{3}: This is a product of two numbers, 8 and 3\sqrt{3}.
  • 12⋅31212 \cdot 3^{\frac{1}{2}}: This is a product of two numbers, 12 and 3123^{\frac{1}{2}}.
  • ÷\div: This is a division operation.

Evaluating the Exponents

The expression contains an exponent, 3123^{\frac{1}{2}}. To evaluate this, we need to find the square root of 3:

312=33^{\frac{1}{2}} = \sqrt{3}

Evaluating the Multiplication

Now that we have evaluated the exponent, we can evaluate the multiplication:

12⋅312=12⋅312 \cdot 3^{\frac{1}{2}} = 12 \cdot \sqrt{3}

Evaluating the Division

Next, we need to evaluate the division operation:

83÷(12⋅3)8 \sqrt{3} \div (12 \cdot \sqrt{3})

To evaluate this, we can use the rule that aa=1\frac{a}{a} = 1:

83123=812\frac{8 \sqrt{3}}{12 \sqrt{3}} = \frac{8}{12}

Simplifying the Fraction

The fraction 812\frac{8}{12} can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

812=23\frac{8}{12} = \frac{2}{3}

Evaluating the Final Expression

Now that we have evaluated the division operation, we can evaluate the final expression:

−1+23-1 + \frac{2}{3}

To evaluate this, we can use the rule that a+bc=ac+bca + \frac{b}{c} = \frac{ac + b}{c}:

−1+23=−3+23=−13-1 + \frac{2}{3} = \frac{-3 + 2}{3} = \frac{-1}{3}

Conclusion

In conclusion, the expression −1+83÷(12⋅312)-1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right) evaluates to −13\frac{-1}{3}.

Final Answer

The final answer is −13\boxed{\frac{-1}{3}}.

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Understanding the Expression

The given expression is −1+83÷(12⋅312)-1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right). To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Breaking Down the Expression

Let's break down the expression into smaller parts to make it easier to evaluate:

  • 838 \sqrt{3}: This is a product of two numbers, 8 and 3\sqrt{3}.
  • 12⋅31212 \cdot 3^{\frac{1}{2}}: This is a product of two numbers, 12 and 3123^{\frac{1}{2}}.
  • ÷\div: This is a division operation.

Evaluating the Exponents

The expression contains an exponent, 3123^{\frac{1}{2}}. To evaluate this, we need to find the square root of 3:

312=33^{\frac{1}{2}} = \sqrt{3}

Evaluating the Multiplication

Now that we have evaluated the exponent, we can evaluate the multiplication:

12⋅312=12⋅312 \cdot 3^{\frac{1}{2}} = 12 \cdot \sqrt{3}

Evaluating the Division

Next, we need to evaluate the division operation:

83÷(12⋅3)8 \sqrt{3} \div (12 \cdot \sqrt{3})

To evaluate this, we can use the rule that aa=1\frac{a}{a} = 1:

83123=812\frac{8 \sqrt{3}}{12 \sqrt{3}} = \frac{8}{12}

Simplifying the Fraction

The fraction 812\frac{8}{12} can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

812=23\frac{8}{12} = \frac{2}{3}

Evaluating the Final Expression

Now that we have evaluated the division operation, we can evaluate the final expression:

−1+23-1 + \frac{2}{3}

To evaluate this, we can use the rule that a+bc=ac+bca + \frac{b}{c} = \frac{ac + b}{c}:

−1+23=−3+23=−13-1 + \frac{2}{3} = \frac{-3 + 2}{3} = \frac{-1}{3}

Conclusion

In conclusion, the expression −1+83÷(12⋅312)-1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right) evaluates to −13\frac{-1}{3}.

Final Answer

The final answer is −13\boxed{\frac{-1}{3}}.

Q&A

Q: What is the order of operations?

A: The order of operations is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.

Q: How do I evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, you need to follow the order of operations and evaluate any exponential expressions first, then any multiplication and division operations, and finally any addition and subtraction operations.

Q: What is the difference between multiplication and division?

A: Multiplication and division are both operations that involve numbers, but they have different rules. Multiplication involves multiplying two numbers together, while division involves dividing one number by another.

Q: How do I simplify fractions?

A: To simplify fractions, you need to divide both the numerator and the denominator by their greatest common divisor.

Q: What is the final answer to the expression −1+83÷(12⋅312)-1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right)?

A: The final answer to the expression −1+83÷(12⋅312)-1 + 8 \sqrt{3} \div \left(12 \cdot 3^{\frac{1}{2}}\right) is −13\frac{-1}{3}.

Q: Can I use a calculator to evaluate the expression?

A: Yes, you can use a calculator to evaluate the expression, but it's also important to understand the steps involved in evaluating the expression to ensure that you get the correct answer.

Q: What if I get a different answer than −13\frac{-1}{3}?

A: If you get a different answer than −13\frac{-1}{3}, it's possible that you made a mistake in evaluating the expression. Double-check your work and make sure that you followed the order of operations correctly.