Question

1. a rectangle has length 8 × 10⁴ millimeters and width 6 × 10⁴ millimeters. how many times greater is the rectangle’s length than width ______ (standard form)

Explanation:

Step1: Set up the division

To find how many times greater the length is than the width, divide the length by the width: $\frac{8\times10^{4}}{6\times10^{4}}$

Step2: Simplify the expression

Using the rule of exponents $\frac{a\times10^{n}}{b\times10^{n}}=\frac{a}{b}$ (since $10^{4}$ cancels out), we get $\frac{8}{6}=\frac{4}{3}\approx1.333\cdots$? Wait, no, wait, maybe I misread. Wait, the length is $8\times10^{4}$ and width is $6\times10^{4}$? Wait, no, maybe the exponents are different? Wait, no, the user's image: "length 8 × 10⁴ millimeters and width 6 × 10⁴ millimeters. How many times greater is the rectangle’s length than width". Wait, no, that can't be, maybe a typo? Wait, maybe length is $8\times10^{4}$ and width is $6\times10^{3}$? Wait, no, the image says "8 × 10⁴" and "6 × 10⁴". Wait, but if that's the case, then $\frac{8\times10^{4}}{6\times10^{4}}=\frac{8}{6}=\frac{4}{3}\approx1.33$. But maybe I misread the exponents. Wait, maybe the width is $6\times10^{3}$? Wait, the user's image: let me check again. The text: "A rectangle has length 8 × 10⁴ millimeters and width 6 × 10⁴ millimeters. How many times greater is the rectangle’s length than width". Wait, that would be $\frac{8\times10^{4}}{6\times10^{4}}=\frac{8}{6}=\frac{4}{3}\approx1.33$. But maybe it's a different exponent. Wait, maybe the width is $6\times10^{3}$? Let me assume that maybe there's a typo, but according to the given, it's $10^{4}$ for both. Wait, no, maybe I made a mistake. Wait, let's do it as per the given.

Wait, the length is $8\times10^{4}$, width is $6\times10^{4}$. So dividing them: $\frac{8\times10^{4}}{6\times10^{4}}=\frac{8}{6}=\frac{4}{3}\approx1.33$. But that seems odd. Wait, maybe the width is $6\times10^{3}$? Let's check the original problem again. The user's image: "length 8 × 10⁴ millimeters and width 6 × 10⁴ millimeters". So exponents are same. So the ratio is 8/6 = 4/3 ≈ 1.33. But maybe the problem was supposed to have different exponents. Wait, maybe the width is $6\times10^{3}$? Let's suppose that. Then length is $8\times10^{4}$, width is $6\times10^{3}$. Then $\frac{8\times10^{4}}{6\times10^{3}}=\frac{8}{6}\times10^{1}=\frac{4}{3}\times10=\frac{40}{3}\approx13.33$. But according to the given, exponents are same. Wait, maybe the original problem has a different exponent. Wait, maybe the user made a typo. But as per the given, let's proceed.

Wait, no, maybe I misread the length and width. Let me check again. The text: "length 8 × 10⁴ millimeters and width 6 × 10⁴ millimeters. How many times greater is the rectangle’s length than width". So the calculation is (8×10⁴) ÷ (6×10⁴) = 8/6 = 4/3 ≈ 1.33. But that seems too simple. Wait, maybe the length is 8×10⁵ and width 6×10⁴? Let's see. Then (8×10⁵)/(6×10⁴) = (8/6)×10 = 40/3 ≈13.33. But according to the given, it's 10⁴ for both. So maybe the answer is 4/3 or 1.33. But let's do it step by step.

Step1: Write the division

We need to find $\frac{\text{Length}}{\text{Width}}=\frac{8\times10^{4}}{6\times10^{4}}$

Step2: Simplify the powers of 10

Using the property of exponents $\frac{10^{m}}{10^{n}}=10^{m - n}$, here m = n = 4, so $10^{4-4}=10^{0}=1$

Step3: Simplify the coefficients

$\frac{8}{6}=\frac{4}{3}\approx1.33$

Step4: Multiply the results

$\frac{4}{3}\times1 = \frac{4}{3}$ or approximately 1.33

Wait, but maybe the problem was intended to have different exponents. For example, if the width was $6\times10^{3}$, then:

Step1: Write the division

$\frac{8\times10^{4}}{6\times10^{3}}$

Step2: Simplify the powers of 10

$10^{4-3}=10^{1}=10$

Step3: Simplify the coefficients

$\frac{8}{6}=\frac{4}{3}$

Step4: Multiply

$\frac{4}{3}\times10=\frac{40}{3}\approx13.33$

But according to the given problem, the exponents are both $10^{4}$. So the answer is $\frac{4}{3}$ or approximately 1.33.

Answer:

$\frac{4}{3}$ (or approximately $1.33$)