Problem 8
Question
Evaluate each expression. Retain the proper number of significant digits in your answer. Powers. $$10^{4}$$
Step-by-Step Solution
Verified Answer
The value of \(10^{4}\) is 10000.
1Step 1: Understanding the Power Expression
The expression given is a power expression of the form 10 to the power of 4, written as \(10^{4}\). This implies multiplying the number 10 by itself a total of 4 times.
2Step 2: Evaluating the Power Expression
To solve the expression, simply perform the multiplication: \(10 \times 10 \times 10 \times 10 = 10000\).
3Step 3: Applying Significant Digits Rules
Since 10 is an exact number in this context, there are an infinite number of significant digits, and the result should be reported as 10000.
Key Concepts
Significant DigitsExponential ExpressionsMathematical Operations
Significant Digits
Understanding significant digits, also known as significant figures, is crucial when working with measurements and expressing accuracy in numerical results. These are the digits in a number that carry meaning contributing to its precision. This includes all non-zero numbers, any zeros between significant digits, and trailing zeros in a decimal number.
For example, in the number 123.45, all digits are significant because they represent actual measured values. However, trailing zeros, as in 1500, can be ambiguous—they can be placeholders or significant if measured or defined.
When performing mathematical operations, the resulting value should have the same number of significant figures as the measurement with the least amount of significant digits. In our power expression example with the number 10 to the power of 4 (\(10^{4}\)), the number 10 is considered to have an infinite number of significant digits because it's an exact count. Therefore, the result of 10000 maintains all the significant digits from the initial exact values.
For example, in the number 123.45, all digits are significant because they represent actual measured values. However, trailing zeros, as in 1500, can be ambiguous—they can be placeholders or significant if measured or defined.
When performing mathematical operations, the resulting value should have the same number of significant figures as the measurement with the least amount of significant digits. In our power expression example with the number 10 to the power of 4 (\(10^{4}\)), the number 10 is considered to have an infinite number of significant digits because it's an exact count. Therefore, the result of 10000 maintains all the significant digits from the initial exact values.
Exponential Expressions
Exponential expressions, like the power expression \(10^{4}\), involve a base number raised to a certain exponent. The exponent dictates how many times the base is multiplied by itself. In simple terms, \(a^{n}\) can be read as 'a' multiplied by itself 'n' times.
These expressions are not limited to positive exponents; they can also be negative, which would imply division, or fractional, which would represent roots. For example, \(10^{-2}\) is the same as 1 divided by \(10^{2}\) or 0.01, and \(8^{\frac{1}{3}}\) denotes the cube root of 8, which is 2.
In mathematics and sciences, exponential expressions are fundamental, often representing growth and decay processes, compound interest, and other phenomena involving rapid change.
These expressions are not limited to positive exponents; they can also be negative, which would imply division, or fractional, which would represent roots. For example, \(10^{-2}\) is the same as 1 divided by \(10^{2}\) or 0.01, and \(8^{\frac{1}{3}}\) denotes the cube root of 8, which is 2.
In mathematics and sciences, exponential expressions are fundamental, often representing growth and decay processes, compound interest, and other phenomena involving rapid change.
Mathematical Operations
Mathematical operations consist of basic processes such as addition, subtraction, multiplication, and division—each with its specific set of rules for handling numbers and their significant digits. Advanced operations like exponents, roots, and logarithms play a significant role in higher-level mathematics and applications.
In the context of evaluating powers, the operation calls for multiplying the base by itself as many times as indicated by the exponent. Correctly performing these basic to advanced mathematical operations is key to solving many algebraic expressions, and understanding the underlying principles helps in recognizing patterns and solving complex mathematical problems.
Always ensure that you follow the correct order of operations, affectionately known as PEMDAS or BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction. This sequence helps to avoid errors and arrive at a proper solution.
In the context of evaluating powers, the operation calls for multiplying the base by itself as many times as indicated by the exponent. Correctly performing these basic to advanced mathematical operations is key to solving many algebraic expressions, and understanding the underlying principles helps in recognizing patterns and solving complex mathematical problems.
Always ensure that you follow the correct order of operations, affectionately known as PEMDAS or BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication, Division, Addition, and Subtraction. This sequence helps to avoid errors and arrive at a proper solution.
Other exercises in this chapter
Problem 8
Convert the following customary units. 8552 ounces to pounds.
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Dividing Approximate Numbers Divide, and then round your answer to the proper number of digits. $$-4.8 \div-2.557$$
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Multiply each approximate number and retain the proper number of digits in your answer. $$4.97 \times 9.27 \times 5.78$$
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Evaluate each expression. $$|-3|$$
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