Power and Root Functions -: Videos & Practice Problems

When manipulating numbers in scientific notation, raising a value to a power involves cubing the coefficient and multiplying the exponent by that power. For example,

$(3.0\times {10}^{-2}){)}^3$

becomes

$2.7\times {10}^{-5}$

after adjusting the coefficient to fall between 1 and 10 and modifying the exponent accordingly. Taking roots, such as the cube root, is akin to raising the number to the reciprocal power (e.g., the cube root is the same as raising to the 1/3 power). This results in adjusting both the coefficient and the exponent by the reciprocal power, yielding a new scientific notation expression. Calculators may have different functions for these operations, such as a root button or using a power button with the reciprocal fraction. Precision in entering these calculations is crucial to ensure the correct answer is obtained.

Power and Root Functions Example

When working with numbers in scientific notation, understanding how to manipulate them through exponentiation and root functions is essential. When raising a number in scientific notation to a power, the coefficient is raised to that power, while the exponent is multiplied by the power. For example, if we take the number \(3.0 \times 10^{-2}\) and cube it, we first cube the coefficient: \(3^3 = 27\). Next, we multiply the exponent by the power, resulting in \(10^{-2 \times 3} = 10^{-6}\). However, since \(27\) is not a valid coefficient in scientific notation (which must be between 1 and 10), we adjust it to \(2.7\) by moving the decimal point one place to the left. This adjustment increases the exponent by 1, leading to the final expression of \(2.7 \times 10^{-5}\).

When taking a root of a number in scientific notation, the process involves raising the coefficient to the reciprocal of the root's degree. For instance, the cube root is equivalent to raising to the power of \(\frac{1}{3}\). If we consider \(6.0 \times 10^{9}\) and take the cube root, we calculate \(6.0^{\frac{1}{3}} \approx 1.8\). Simultaneously, we multiply the exponent by \(\frac{1}{3}\), resulting in \(10^{9 \times \frac{1}{3}} = 10^{3}\). Thus, the final result is \(1.8 \times 10^{3}\).

To perform these calculations on a calculator, you may need to use specific functions. For cube roots, look for a button that resembles a square root with an 'x' or a caret (^) symbol. If your calculator does not have a direct cube root function, you can raise the number to the power of \(\frac{1}{3}\) by entering the number in parentheses, followed by the caret symbol, and then \(\frac{1}{3}\) in parentheses. This method will yield the same result.

Practicing these operations will enhance your proficiency in handling scientific notation, ensuring accurate calculations whether you are raising numbers to powers or extracting roots.

Power and Root Functions Example

To solve the expression \( (7.5 \times 10^{-3})^5 \times \sqrt[4]{8.6 \times 10^{21}} \), we start by calculating each part separately. First, raise \( 7.5 \times 10^{-3} \) to the 5th power. This can be computed as follows:

Using the power of a product rule, we have:

\[(7.5 \times 10^{-3})^5 = 7.5^5 \times (10^{-3})^5 = 23730.0005 \times 10^{-15} = 2.3730 \times 10^{-11}\]

Next, we need to find the 4th root of \( 8.6 \times 10^{21} \). The 4th root can be expressed as raising to the power of \( \frac{1}{4} \):

\[\sqrt[4]{8.6 \times 10^{21}} = (8.6 \times 10^{21})^{\frac{1}{4}} = 8.6^{\frac{1}{4}} \times (10^{21})^{\frac{1}{4}} = 1.903 \times 10^{5.25} \approx 304526\]

Now, we multiply the results of these two calculations:

\[(2.3730 \times 10^{-11}) \times (304526) \approx 7.2 \times 10^{-6}\]

When performing multiplication, it is essential to consider significant figures. The coefficient \( 2.3730 \) has 4 significant figures, while \( 304526 \) has 6 significant figures. Therefore, the final answer should be reported with 2 significant figures, resulting in:

\[7.2 \times 10^{-6}\]

To ensure accuracy, it is crucial to practice entering these values correctly into your calculator. Consistent practice will help you become more proficient in handling scientific notation and significant figures.

Here’s what students ask on this topic:

How do you raise a number in scientific notation to a power?

To raise a number in scientific notation to a power, you need to raise the coefficient to that power and multiply the exponent by the power. For example, if you have ${3.0}^{10-2}$ and you want to cube it, you would calculate ${3.0}^3$ and $10-2*\; 3$. This results in $27\; *\; 10-6$. Since 27 is not between 1 and 10, adjust it to 2.7 and increase the exponent by 1, giving you $2.7\; *\; 10-5$.

What is the process for taking the cube root of a number in scientific notation?

Taking the cube root of a number in scientific notation involves raising the coefficient to the reciprocal power (1/3) and multiplying the exponent by 1/3. For example, for $6.0\; *\; 109$, you calculate ${6.0}^{1/3}$ and $109*\; 1/3$. This results in $1.8\; *\; 103$. Ensure you use the correct calculator functions to get accurate results.

How do you adjust the coefficient in scientific notation to fall between 1 and 10?

To adjust the coefficient in scientific notation to fall between 1 and 10, move the decimal point in the coefficient and adjust the exponent accordingly. If you make the coefficient smaller, increase the exponent by the number of places you moved the decimal. For example, if you have 27 * 10^-6, move the decimal one place to get 2.7 and increase the exponent by 1, resulting in 2.7 * 10^-5.

What are the different calculator functions for raising numbers to powers and taking roots?

Calculators may have different functions for raising numbers to powers and taking roots. Some calculators have a root button (√) or a power button (x^y or y^x). To take the cube root, you might use the root button with the number 3, or use the power button with the reciprocal fraction (1/3). Ensure you input the values correctly to get accurate results.

Why is precision important when entering calculations into a calculator?

Precision is crucial when entering calculations into a calculator because incorrect input can lead to wrong answers. Understanding how to use the calculator functions properly ensures that you get accurate results, especially when dealing with scientific notation and complex operations like raising to powers or taking roots. Always double-check your inputs and follow the correct steps for your specific calculator model.