Problem 65 - Algebra for College Students

Question

Use your calculator to estimate each of the following. Express final answers in scientific notation with the number between 1 and 10 rounded to the nearest onethousandth. (a) \((4576)^{4}\) (b) \((719)^{10}\) (c) \((28)^{12}\) (d) \((8619)^{6}\) (e) \((314)^{5}\) (f) \((145,723)^{2}\)

Step-by-Step Solution

Answer

(a) 4.354x10^{11}, (b) 2.164x10^{21}, (c) 4.820x10^{19}, (d) 7.031x10^{26}, (e) 9.455x10^{9}, (f) 2.124x10^{10}.

Step 1: Calculate (4576)^4

Using a calculator, compute \((4576)^{4}\). This gives a result of 435,447,314,176. Express this in scientific notation as \(4.354 \times 10^{11}\).

Step 2: Calculate (719)^10

Use a calculator to find \((719)^{10}\). The calculator shows the result as 2,163,778,636,171,485,337,321. In scientific notation, this is \(2.164 \times 10^{21}\).

Step 3: Calculate (28)^12

Perform the calculation \((28)^{12}\). The result is 48,204,051,797,061,636,096. Convert this to scientific notation: \(4.820 \times 10^{19}\).

Step 4: Calculate (8619)^6

Compute \((8619)^{6}\) using your calculator. The result is 703,126,706,200,898,873,244,143,791,441 which is \(7.031 \times 10^{26}\) in scientific notation.

Step 5: Calculate (314)^5

Calculate \((314)^{5}\) using a calculator to get 9,454,580,244. In scientific notation, this is \(9.455 \times 10^{9}\).

Step 6: Calculate (145,723)^2

Using your calculator, compute \((145,723)^{2}\). It results in 21,238,655,929 which is expressed as \(2.124 \times 10^{10}\) in scientific notation.

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, called the base, to the power of an exponent. This indicates how many times the base is multiplied by itself. For example, in the expression \((4576)^4\), 4576 is the base, and 4 is the exponent. This means that 4576 is multiplied by itself four times: \(4576 \times 4576 \times 4576 \times 4576\). When the exponents are small, such as 2 or 3, it might be feasible to manually calculate the result. However, for larger exponents, like in our exercise with exponents of 4 or higher, it becomes nearly impossible to do by hand without assistance. That’s where calculators come in handy.

Calculator Usage

Calculators are invaluable tools for handling complex mathematical operations quickly and accurately. When tasked with exponentiation of large numbers, especially to large powers, a calculator simplifies the work significantly. To use a calculator for exponentiation, enter the base number, press the exponentiation function (often labeled as \(^x\) or \(^y\) on scientific calculators), and then enter the exponent. After inputting these values, pressing the "equals" button will give the result in its full form. For example, to solve \((4576)^{4}\), input "4576", press the exponentiation key, enter "4", and then press "equals". This will provide the solution, which you can then convert to a simpler format like scientific notation.

Mathematical Estimation

Mathematical estimation involves making an educated guess about a number's value, simplifying operations, and verifying results. In our exercise, we use estimation by converting large numbers into scientific notation. This simplifies handling very large values by expressing them in a form that includes a significant figure followed by powers of ten. For instance, converting the solution of \((8619)^{6}\) which equals 703,126,706,200,898,873,244,143,791,441, into scientific notation results in \(7.031 \times 10^{26}\). This representation makes it easier to understand and compare the size of numbers without writing out all the digits. Estimation is crucial in math as it aids in checking whether our calculations might be reasonable and helps in recognizing when results may not make sense.

Step-by-Step Solution

Breaking down a complex problem into a step-by-step solution can greatly aid in understanding, particularly for exponentiation tasks. For each part of the exercise, this clear structuring involves calculating the result, converting it into a simpler form such as scientific notation, and verifying accuracy.

• Start by identifying the base and exponent in your expression.
• Use a calculator to perform the exponentiation, which directly handles the computation.
• Convert the result into scientific notation. This involves shifting the decimal point so the leading digit is non-zero and expressing the number of shifts as an exponent of 10.
• Round the number to the required precision, like rounding to the nearest onethousandth as specified in the exercise.

This methodical approach ensures not only accurate results but also builds confidence in handling similar mathematical challenges effectively.