Using laws of indices, simplify the following:
(a) $3 x^4 \times x^2$
Answer (2)
Apply the rule of indices: x m × x n = x m + n .
Rewrite the expression as 3 × ( x 4 × x 2 ) .
Add the exponents: 4 + 2 = 6 .
The simplified expression is 3 x 6 .
Explanation
Understanding the Problem We are asked to simplify the expression 3 x 4 × x 2 using the laws of indices. These laws tell us how to manipulate exponents when multiplying terms with the same base.
Applying the Rule of Indices The key rule we'll use is: x m × x n = x m + n . This means when you multiply two terms with the same base (in this case, x ), you add their exponents.
Rewriting the Expression Let's rewrite the expression to make it clearer: 3 x 4 × x 2 = 3 × ( x 4 × x 2 ) . Now, we focus on the terms with the base x .
Adding the Exponents Adding the exponents, we have x 4 × x 2 = x 4 + 2 = x 6 . So, the expression becomes 3 × x 6 , which is simply 3 x 6 .
Final Answer Therefore, the simplified expression is 3 x 6 .
Examples
Understanding and simplifying expressions using laws of indices is crucial in many areas, such as calculating areas and volumes. For instance, if you are calculating the volume of a rectangular prism with sides x 2 , 3 x 4 , and x , the volume would be x 2 × 3 x 4 × x = 3 x 2 + 4 + 1 = 3 x 7 . Simplifying expressions helps in making these calculations easier and more manageable.
To simplify 3 x 4 × x 2 , we apply the law of indices which states that x m × x n = x m + n . By adding the exponents 4 + 2 , we get x 6 , leading to the final simplified expression of 3 x 6 .
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