Question

what is the simplified form of the following expression? 10abc² √(49a²b²c²) options: 7|abc|, 7a²b²|c³|, 70|abc|c², 70abc²|abc|

Explanation:

Step1: Simplify the square root term

First, simplify $\sqrt{49a^2b^2c}$. We know that $\sqrt{49}=7$, $\sqrt{a^2}=|a|$, $\sqrt{b^2}=|b|$, and $\sqrt{c}$ remains as is (but since we have $a^2$ and $b^2$, their square roots are absolute values, but when multiplied with other terms, let's proceed). So $\sqrt{49a^2b^2c} = 7|a||b|\sqrt{c}$. Wait, but maybe the original expression is $10abc^2 \times \sqrt{49a^2b^2c}$? Wait, the first term is $10abc^2$ and then times $\sqrt{49a^2b^2c}$? Wait, let's re - examine. Wait, maybe the expression is $10abc^2 \cdot \sqrt{49a^2b^2c}$. Let's compute the square root first: $\sqrt{49a^2b^2c}=\sqrt{49}\cdot\sqrt{a^2}\cdot\sqrt{b^2}\cdot\sqrt{c}=7|a||b|\sqrt{c}$. But maybe in the context of the problem, we assume $a,b$ are non - negative (since it's a simplification problem, maybe we can drop the absolute values for $a$ and $b$ if we assume they are non - negative, or maybe there's a typo). Wait, maybe the original expression is $10abc^2\times\sqrt{49a^2b^2c}$. Let's compute the numerical coefficient and the variable coefficients separately.

The numerical part: $10\times7 = 70$.

For the variable $a$: $a\times a=a^{1 + 1}=a^2$ (since $\sqrt{a^2}=a$ if $a\geq0$).

For the variable $b$: $b\times b = b^{1+1}=b^2$ (since $\sqrt{b^2}=b$ if $b\geq0$).

For the variable $c$: $c^2\times\sqrt{c}=c^2\times c^{\frac{1}{2}}=c^{2+\frac{1}{2}}=c^{\frac{5}{2}}$? Wait, no, maybe I misread the original expression. Wait, the original expression is $10abc^2\sqrt{49a^2b^2c}$. Wait, maybe the exponent of $c$ in the square root is 1, and in the first term is 2. Let's do the multiplication of the variables:

For $a$: $a\times a = a^{2}$ (because $\sqrt{a^2}=a$)

For $b$: $b\times b = b^{2}$ (because $\sqrt{b^2}=b$)

For $c$: $c^2\times\sqrt{c}=c^2\times c^{\frac{1}{2}}=c^{\frac{5}{2}}$? No, that doesn't match the options. Wait, maybe the original expression is $10abc^2\times\sqrt{49a^2b^2c}$, and we made a mistake. Wait, the options have $70abc^2|abc|$? Wait, the last option is $70abc^2|abc|$? Wait, no, the last option is $70abc^2|abc|$? Wait, looking at the options:

Option 1: $7|abc|$

Option 2: $7a^2b^2|c|$

Option 3: $70|abc|c^2$? Wait, no, the third option is $70|abc|c^2$? Wait, the user's image shows options:

1. $7|abc|$

2. $7a^2b^2|c|$

3. $70|abc|c^2$? Wait, no, the third option is $70|abc|c^2$? Wait, the first term is $10abc^2$, the square root term is $\sqrt{49a^2b^2c}=7|a||b|\sqrt{c}$. Then multiplying $10abc^2\times7|a||b|\sqrt{c}=70|a|^2|b|^2c^2\sqrt{c}$. But $|a|^2 = a^2$, $|b|^2 = b^2$, so it becomes $70a^2b^2c^2\sqrt{c}$? No, that's not matching. Wait, maybe the original expression is $10abc^2\times\sqrt{49a^2b^2c^3}$? Wait, maybe there's a typo in the exponent of $c$ in the square root. If the square root is $\sqrt{49a^2b^2c^3}$, then $\sqrt{c^3}=c\sqrt{c}$, so $\sqrt{49a^2b^2c^3}=7|a||b|c\sqrt{c}$. Then multiplying with $10abc^2$: $10\times7 = 70$, $a\times|a|$, $b\times|b|$, $c^2\times c=c^3$, and $\sqrt{c}$. No, this is getting confusing. Wait, let's look at the options. The fourth option is $70abc^2|abc|$. Let's check the numerical coefficient: $10\times7 = 70$. For the variables: $a\times a=a^2$, $b\times b = b^2$, $c^2\times c=c^3$? No, wait, if we consider $|abc|=|a||b||c|$, and $abc^2\times|abc|=abc^2\times|a||b||c|$. But if we assume $a,b$ are non - negative, then $|a| = a$, $|b|=b$, so $abc^2\times abc= a^{2}b^{2}c^{3}$, and then multiply by 7 (from the square root) and 10? Wait, no, the square root of 49 is 7, and the first term is 10. So $10\times7 = 70$. Then for the variables: $a$ (from first term) times $a$ (from square root of $a^2$) is $a^2$, $b$ times $b$ is $b^2$, $c^2$ times $c$ (from square root of $c$? No, maybe the square root is $\sqrt{49a^2b^2c}=7ab\sqrt{c}$ (assuming $a,b\geq0$). Then $10abc^2\times7ab\sqrt{c}=70a^{2}b^{2}c^{2}\sqrt{c}$. But this is not matching the options. Wait, maybe the original expression is $10abc^2\times\sqrt{49a^2b^2c}=10abc^2\times7ab\sqrt{c}=70a^{2}b^{2}c^{2}\sqrt{c}$. But the options don't have this. Wait, maybe the question is $10abc^2\times\sqrt{49a^2b^2c^2}$? If the square root is $\sqrt{49a^2b^2c^2}$, then $\sqrt{49a^2b^2c^2}=7|a||b||c|$. Then $10abc^2\times7|a||b||c|=70abc^2|abc|$. Ah, that makes sense! If the square root is of $49a^2b^2c^2$ (maybe a typo in the exponent of $c$), then $\sqrt{49a^2b^2c^2}=7|a||b||c|$. Then multiplying with $10abc^2$:

Numerical coefficient: $10\times7 = 70$

Variable part: $abc^2\times|a||b||c|=abc^2|abc|$ (since $|a||b||c| = |abc|$)

So the product is $70abc^2|abc|$, which is the fourth option.

Step2: Verify the options

Looking at the options, the fourth option is $70abc^2|abc|$, which matches our calculation when we assume the square root is of $49a^2b^2c^2$ (maybe a typo in the original problem's exponent of $c$ in the square root term).

Answer:

The correct option is the fourth one: $70abc^2|abc|$