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Evaluate $\\frac{xy}{z}$ with Standard Form Values and Return Answer in Standard Form
Mathematics
Grade 9 (Junior High School)
Question Content
Calculate $\frac{xy}{z}$ using the values below. $x = 3 \\times 10^4$, $y = 4 \\times 10^{10}$, $z = 2 \\times 10^2$. Give your answer in standard form.
Correct Answer
$6 \\times 10^{12}$
Detailed Solution Steps
1
Step 1: Substitute the given values into the expression: $\frac{xy}{z} = \\frac{(3 \\times 10^4) \\times (4 \\times 10^{10})}{2 \\times 10^2}$
2
Step 2: Separate the coefficient terms and the power-of-10 terms: $\\frac{3 \\times 4}{2} \\times \\frac{10^4 \\times 10^{10}}{10^2}$
3
Step 3: Calculate the coefficient part: $\\frac{12}{2} = 6$
4
Step 4: Use exponent rules to simplify the power-of-10 part: For multiplication of exponents with the same base, add the exponents ($10^4 \\times 10^{10} = 10^{4+10}=10^{14}$); for division, subtract the exponents ($\\frac{10^{14}}{10^2}=10^{14-2}=10^{12}$)
5
Step 5: Combine the results to get the final answer in standard form: $6 \\times 10^{12}$
Knowledge Points Involved
1
Standard Form (Scientific Notation)
Standard form is a way to write very large or very small numbers as $a \\times 10^n$, where $1 \\leq |a| < 10$ and $n$ is an integer. It is used to simplify calculations with extreme values and make them easier to read and compare.
2
Exponent Rule for Multiplication
For exponents with the same base $b$, $b^m \\times b^n = b^{m+n}$. This rule allows us to combine the powers of 10 when multiplying numbers in standard form by adding their exponents instead of expanding the full numbers.
3
Exponent Rule for Division
For exponents with the same base $b$, $\\frac{b^m}{b^n} = b^{m-n}$. This rule simplifies dividing numbers in standard form by subtracting the exponent of the denominator from the exponent of the numerator.
4
Algebraic Substitution
This is the process of replacing variables in an expression with their given numerical values. It is a foundational step for evaluating algebraic expressions with known variable values.
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