Math  /  Algebra

Question7. What is the solution 25x=410\frac{2}{5} x=\frac{4}{10} ? A) 10 B) 5 C) 4 D) 1
8. What is the simplest form of b10b4\frac{b^{10}}{b^{4}} ? A) b6b^{6} B) b6b^{-6} C) b14b^{14} D) 1b6\frac{1}{b^{6}}
9. What is the simplest form of x3?x2xxx\sqrt{x^{3}} ? \sqrt{x^{2} x}-x \sqrt{x} A) x2xx^{2} \sqrt{x} B) xxx \sqrt{x} C) xx2x \sqrt{x^{2}} D) 3x3 \sqrt{x}
10. What is the simplest form of the expression 75×3\sqrt{75} \times \sqrt{3} ? A) 15\sqrt{15} B) 535 \sqrt{3} C) 15 D) 225

Studdy Solution

STEP 1

What is this asking? We've got a few separate problems here: solve for xx in an equation, simplify an exponent expression, simplify a square root expression, and simplify a product of square roots. Watch out! Remember the rules of fractions and exponents!
Don't mix them up, and be careful with those square roots.

STEP 2

1. Solve for *x*
2. Simplify the exponent expression
3. Simplify the first square root expression
4. Simplify the product of square roots

STEP 3

We're asked to solve 25x=410\frac{2}{5}x = \frac{4}{10}.
Let's **isolate** xx by multiplying both sides by 52\frac{5}{2}.
Remember, we do this because 5225=1\frac{5}{2} \cdot \frac{2}{5} = 1, and multiplying by one doesn't change the value!

STEP 4

5225x=41052 \frac{5}{2} \cdot \frac{2}{5}x = \frac{4}{10} \cdot \frac{5}{2}

STEP 5

Now, let's simplify.
On the left side, we have 1x1 \cdot x, which is just xx.
Awesome! On the right side, we have 45102=2020\frac{4 \cdot 5}{10 \cdot 2} = \frac{20}{20}.

STEP 6

So, x=2020=1x = \frac{20}{20} = 1. **Final Answer:** x=1x = 1.

STEP 7

We have b10b4\frac{b^{10}}{b^4}.
When dividing terms with the same base, we *subtract* the exponents.

STEP 8

b10b4=b104=b6 \frac{b^{10}}{b^4} = b^{10-4} = b^6 **Final Answer:** b6b^6.

STEP 9

We're looking at x3\sqrt{x^3}.
We can rewrite x3x^3 as x2xx^2 \cdot x.

STEP 10

x3=x2x \sqrt{x^3} = \sqrt{x^2 \cdot x}

STEP 11

Using the property ab=ab\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}, we get: x2x=x2x \sqrt{x^2 \cdot x} = \sqrt{x^2} \cdot \sqrt{x}

STEP 12

Since x2=x\sqrt{x^2} = x, we have xxx\sqrt{x}. **Final Answer:** xxx\sqrt{x}.

STEP 13

We have 75×3\sqrt{75} \times \sqrt{3}.
We can rewrite this as 753\sqrt{75 \cdot 3}.

STEP 14

753=225 \sqrt{75 \cdot 3} = \sqrt{225}

STEP 15

Since 1515=22515 \cdot 15 = 225, we have 225=15\sqrt{225} = 15. **Final Answer:** 1515.

STEP 16

Problem 7: D) 1 Problem 8: A) b6b^6 Problem 9: B) xxx\sqrt{x} Problem 10: C) 15

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