Problem 71 - Elementary Algebra

Question

Perform the following operations. $$ \left(5.9 \times 10^{14}\right)^{2} $$

Step-by-Step Solution

Answer

Answer: The square of the number 5.9 x 10^14 is 34.81 x 10^28 in scientific notation.

Step 1: Square the mantissa

First, square the mantissa part (5.9) of the given number: $$ (5.9)^{2} = 34.81 $$

Step 2: Square the exponent

Now, square the exponent part (10^14) of the given number: $$ \left(10^{14}\right)^{2} = 10^{14\times2} = 10^{28} $$

Step 3: Multiply both parts

Finally, multiply both results obtained in Steps 1 and 2: $$ 34.81 \times 10^{28} $$ This is the final result of the given operation.

Key Concepts

Squaring Numbers

Squaring a number means multiplying that number by itself. It's denoted as raising a number to the power of two, written as an exponent. For instance, squaring the number 5, symbolically represented as \(5^2\), results in \(25\), because \(5 \times 5 = 25\).

In the context of the given problem, squaring the mantissa 5.9 means multiplying 5.9 by itself, resulting in \(34.81\). This is a crucial step in finding the square of a number in scientific notation because the mantissa carries the significant digits of the number.

Exponentiation

Exponentiation is a mathematical operation where a number (the base) is raised to the power of an exponent. The exponent tells you how many times to multiply the base by itself. For example, \(2^3\) (read as 'two to the third power' or 'two cubed') is equal to \(2 \times 2 \times 2 = 8\).

In our exercise, exponentiation comes into play when squaring the exponent part of the scientific notation. Squaring the exponent essentially means doubling it, as seen with \(10^{14}\) becoming \(10^{28}\) after squaring because \(14 \times 2 = 28\). This operation is fundamental to performing calculations with exponents.

Mantissa

In scientific notation, the mantissa is the significant figure or the decimal part that contains the precision of the number. Scientific notation is used to express very large or very small numbers concisely, typically in the format of a mantissa multiplied by 10 raised to an exponent. For example, in \(3.16 \times 10^5\), \(3.16\) is the mantissa.

When operations are performed on numbers in scientific notation, the mantissa is manipulated separately from the exponent. As in our example, the mantissa 5.9 is dealt with before addressing the \(10^{14}\) component. Ensuring the accuracy of the mantissa is important because it directly affects the overall value of the number.

Elementary Algebra

Elementary algebra encompasses the fundamental concepts of algebra, including operations on numbers with unknown variables, use of symbols, and the manipulation of algebraic expressions and equations. It's the springboard into more complex areas of mathematics and is essential for solving problems across sciences.

In terms of scientific notation, algebraic rules are applied when squaring the number, as seen with \((5.9 \times 10^{14})^2\). Here, elementary algebra governs the operations as the mantissa and the exponent are handled separately following algebraic principles - first squaring the mantissa 5.9, then squaring the exponent \(10^{14}\), and finally multiplying the results to obtain \(34.81 \times 10^{28}\).