Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Fractional and Demical Exponents. $$(1.07)^{1 / 5}$$

Short answer

The evaluated expression is approximately 1.01 when rounded to three significant digits.

Step-by-step solution

Understand the Concept of Significant Digits

Before you evaluate the expression, it's important to understand that significant digits (or significant figures) are the digits in a number that carry meaning contributing to its precision. This includes all the non-zero digits, any zeros between non-zero digits, and any trailing zeros in the decimal part.

Evaluate the Exponent

To evaluate the expression with fractional exponents, remember that for a number a and a fraction p/q, the expression a^(p/q) represents the q-th root of a raised to the power of p. In this case, you will calculate the fifth root of 1.07.

Use a Calculator

Use a calculator to find the 5th root of 1.07. The result should be displayed with the proper number of significant digits based on the original number, which has three significant digits.

Round to the Proper Number of Significant Digits

If necessary, round the result from the calculator to three significant digits, because the base 1.07 has three significant figures.

Key concepts

Understanding Fractional Exponents

Handling fractional exponents can sometimes be tricky, as they require a certain understanding of both exponents and radicals. A fractional exponent, such as the expression \(1.07^{1 / 5}\), represents a combination of taking a root and raising to a power, encapsulated in one operation. In general, for a positive number \(a\) and a fraction \(\frac{p}{q}\), the expression \(a^{\frac{p}{q}}\) is equivalent to taking the q-th root of \(a\) and then raising the result to the power of \(p\).\
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In the given example, \(1.07^{1 / 5}\) means you are looking for the 5th root of 1.07. Since the exponent is 1, you won't have to raise this root to any further power. Understanding this relationship is critical to evaluating expressions with fractional exponents properly and efficiently.\
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Moreover, when dealing with significant figures, one must consider the precision of the original number. Remember, all nonzero digits, zeros between nonzero digits, and trailing zeros in the decimal part are significant. Therefore, since 1.07 has three significant figures, our result should also reflect that level of precision after computing the fractional exponent.

Deciphering Decimal Exponents

Decimal exponents, like fractional exponents, serve as a shorthand for expressing operations involving roots and powers. When an exponent is written in decimal form, it may not be readily apparent what mathematical operation it corresponds to, as opposed to fractional exponents which clearly indicate a root is to be taken.\
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For instance, an expression like \(10^{1.2}\) may seem daunting at first glance. However, the decimal exponent can be converted into a fraction to make the operations clearer. To do this, you can express the decimal as a fraction with a numerator and denominator—1.2 is the same as \(\frac{6}{5}\). Therefore, \(10^{1.2}\) can be rewritten as \(10^{\frac{6}{5}}\), meaning you would take the 5th root of 10 and raise it to the 6th power. Understanding this can make calculating with decimal exponents far less intimidating and more approachable.

Calculating Significant Figures

Accurately calculating significant figures is pivotal to maintaining the integrity of precision in mathematical operations, especially in science and engineering. When performing calculations, one must keep track of the number of significant figures to ensure the answer reflects the correct level of precision based on the input values.\
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For example, with the expression \(1.07^{1 / 5}\), the number 1.07 is the base and has three significant figures. This is because the 1 and the 7 are non-zero digits, and the 0 is between them. So, after finding the fifth root of 1.07 using a calculator, it's essential to round the result to three significant figures to match the precision of the original number. This often involves rounding up or down, depending on the digit following the third significant figure. If the digit is 5 or greater, round up, if it's less than 5, round down.\
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For instance, if the calculator displays 1.01329 as the fifth root, you would round to 1.01, which has three significant figures, to align with the precision of the base number 1.07. It's critical to present your final answer with the correct number of significant figures, as it directly impacts the value’s reliability and accuracy.