Algebra 1 Worksheets | Dynamically Created Algebra 1 Worksheets - Free Printable

We are given a worksheet titled “Multiplying Radical Expressions” with 10 problems to simplify. I will solve each one step-by-step, showing the process and final simplified form.

---

General Rules Used:

- Product Rule: √a · √b = √(a·b)
- Distributive Property: a(b + c) = ab + ac
- FOIL Method: (a + b)(c + d) = ac + ad + bc + bd
- Simplify radicals by factoring out perfect squares.
- Combine like terms when possible.

---

## Problem 1:
5√(45b) · 6√(48b)

Step 1: Multiply coefficients: 5 × 6 = 30
Step 2: Multiply radicals: √(45b) · √(48b) = √[(45b)(48b)] = √(2160b²)

Step 3: Simplify √(2160b²):
Factor 2160:
2160 = 144 × 15 → √(144 × 15 × b²) = 12b√15

So total: 30 × 12b√15 = 360b√15

✔ Answer: 360b√15

---

## Problem 2:
-√(45c) · (-√(12c²) + √(12c³))

Step 1: Distribute -√(45c) over the parentheses:

= (-√(45c)) · (-√(12c²)) + (-√(45c)) · (√(12c³))

= √(45c · 12c²) - √(45c · 12c³)

Step 2: Simplify each term:

First term: √(540c³)
540 = 36 × 15 → √(36 × 15 × c² × c) = 6c√(15c)

Second term: √(540c⁴)
c⁴ is perfect square → √(540c⁴) = c²√540 = c²·6√15 = 6c²√15

Wait — let’s recalculate carefully:

Actually:

First product: √(45c * 12c²) = √(540c³)
540 = 36 × 15 → √(36×15×c²×c) = 6c√(15c)

Second product: √(45c * 12c³) = √(540c⁴) = √(36×15×c⁴) = 6c²√15

So expression becomes:

6c√(15c) - 6c²√15

But note: these are not like terms — one has √(15c), other has √15.

So we leave as:

✔ Answer: 6c√(15c) - 6c²√15

*(Alternatively, factor 6c: 6c[√(15c) - c√15], but not simpler.)*

---

## Problem 3:
(7√(3d²) + 2√5)(4√(5d²) - 5√5)

Note: √(3d²) = d√3 (assuming d ≥ 0)
Similarly, √(5d²) = d√5

So rewrite:

= (7d√3 + 2√5)(4d√5 - 5√5)

Now use FOIL:

First: 7d√3 · 4d√5 = 28d²√15
Outer: 7d√3 · (-5√5) = -35d√15
Inner: 2√5 · 4d√5 = 8d · 5 = 40d
Last: 2√5 · (-5√5) = -10 · 5 = -50

Combine:

28d²√15 - 35d√15 + 40d - 50

Group like terms:

Terms with √15: d√15 (28d - 35)
Constants: 40d - 50

✔ Answer: 28d²√15 - 35d√15 + 40d - 50

*(Can also write as: d√15(28d - 35) + 10(4d - 5), but not required.)*

---

## Problem 4:
(-3√(7g²) - 7√2)(6√(7g²) + √2)

Again, √(7g²) = g√7 (assume g ≥ 0)

Rewrite:

= (-3g√7 - 7√2)(6g√7 + √2)

Use FOIL:

First: (-3g√7)(6g√7) = -18g²·7 = -126g²
Outer: (-3g√7)(√2) = -3g√14
Inner: (-7√2)(6g√7) = -42g√14
Last: (-7√2)(√2) = -7·2 = -14

Combine:

-126g² - 3g√14 - 42g√14 - 14
= -126g² - 45g√14 - 14

✔ Answer: -126g² - 45g√14 - 14

---

## Problem 5:
-4√63 (5√48 + √6)

First, simplify radicals inside:

√63 = √(9×7) = 3√7
√48 = √(16×3) = 4√3
√6 remains

So expression becomes:

-4·3√7 · (5·4√3 + √6) = -12√7 (20√3 + √6)

Now distribute:

= -12√7 · 20√3 + (-12√7) · √6
= -240√21 - 12√42

✔ Answer: -240√21 - 12√42

---

## Problem 6:
-6√8 · 7√63

Multiply coefficients: -6 × 7 = -42
Multiply radicals: √8 · √63 = √(8×63) = √504

Simplify √504:
504 = 36 × 14 → √(36×14) = 6√14

Total: -42 × 6√14 = -252√14

✔ Answer: -252√14

---

## Problem 7:
√(32n) (√(20n²) - √(99n³))

Simplify each radical:

√(32n) = √(16×2n) = 4√(2n)
√(20n²) = √(4×5×n²) = 2n√5
√(99n³) = √(9×11×n²×n) = 3n√(11n)

So expression:

4√(2n) [2n√5 - 3n√(11n)]

Distribute:

= 4√(2n) · 2n√5 - 4√(2n) · 3n√(11n)
= 8n √(10n) - 12n √(22n²)

Simplify second term: √(22n²) = n√22

So: 8n√(10n) - 12n · n√22 = 8n√(10n) - 12n²√22

✔ Answer: 8n√(10n) - 12n²√22

---

## Problem 8:
-√(112p) · √(44p)

Multiply radicals: -√(112p · 44p) = -√(4928p²)

Simplify:

First, 112 × 44 = ?
112 × 44 = (100+12)(40+4) = 100×40 + 100×4 + 12×40 + 12×4 = 4000 + 400 + 480 + 48 = 4928

Now factor 4928:

4928 ÷ 16 = 308
308 ÷ 4 = 77 → so 4928 = 16 × 4 × 77 = 64 × 77

Thus: √(4928p²) = √(64 × 77 × p²) = 8p√77

So answer: -8p√77

✔ Answer: -8p√77

---

## Problem 9:
(-2√3 + 6√11)(3√3 - 6√11)

This is of the form (a + b)(c + d), but notice it's similar to difference of squares if rearranged.

Let me compute directly using FOIL:

First: (-2√3)(3√3) = -6 × 3 = -18
Outer: (-2√3)(-6√11) = 12√33
Inner: (6√11)(3√3) = 18√33
Last: (6√11)(-6√11) = -36 × 11 = -396

Combine:

-18 + 12√33 + 18√33 - 396
= (-18 - 396) + (12 + 18)√33
= -414 + 30√33

✔ Answer: -414 + 30√33

---

## Problem 10:
(-√7 + √11)(-√7 + √11)

This is a square: (a + b)² where a = -√7, b = √11

Or better: (x)² where x = -√7 + √11

So: (-√7 + √11)² = (-√7)² + 2(-√7)(√11) + (√11)²
= 7 - 2√77 + 11
= 18 - 2√77

✔ Answer: 18 - 2√77

---

## ✔ Final Answers Summary:

1) 360b√15
2) 6c√(15c) - 6c²√15
3) 28d²√15 - 35d√15 + 40d - 50
4) -126g² - 45g√14 - 14
5) -240√21 - 12√42
6) -252√14
7) 8n√(10n) - 12n²√22
8) -8p√77
9) -414 + 30√33
10) 18 - 2√77

---

Let me know if you’d like these boxed or formatted for printing!