Multiplying Radicals
Multiply radical expressions (a·ⁿsqrtb x c·ᵐsqrtd) and get the simplified radical form plus decimal result. Free online radicals calculator with step-by-step product for algebra students.
About This Calculator
This calculator multiplies two radical expressions and shows the result in both simplified radical form and decimal form. Each radical expression a·ⁿsqrtb consists of a coefficient (a), an index (n), and a radicand (b) -- for example, 2·^3sqrt5 represents 2 multiplied by the cube root of 5. Multiplying radicals is a fundamental skill in algebra and precalculus that appears across geometry, physics, and engineering applications.
When multiplying radicals with the same index, use the product rule: a·ⁿsqrtb x c·ⁿsqrtd = (axc)·ⁿsqrt(bxd). Multiply the coefficients together, multiply the radicands together, and keep the common index. Then simplify the resulting radical by extracting any perfect nth powers from under the radical sign. For example, 2sqrt8 x 3sqrt2 = 6sqrt16 = 6x4 = 24. When radicals have different indices, this calculator finds the least common multiple (LCM) of the two indices, converts each radical to exponential form with a common denominator, and computes the product. For instance, sqrt2 x ^3sqrt4 = 2^(1/2) x 4^(1/3) = 2^(3/6) x 4^(2/6) = ^6sqrt128 = 2·^6sqrt2.
Step-by-step process: First, multiply the coefficients (a x c). Then, find the common index using LCM. Calculate the combined radicand as b^(commonIndex/index1) x d^(commonIndex/index2). Finally, simplify by extracting perfect nth power factors from the combined radicand. The simplified result displays both the radical form (with any remaining radicand inside the root) and the decimal approximation to 6 decimal places.
Frequently Asked Questions
How do you multiply radicals?
To multiply radicals with the same index, multiply the coefficients together, multiply the radicands together, and keep the common index: a·ⁿsqrtb x c·ⁿsqrtd = (axc)·ⁿsqrt(bxd). For radicals with different indices, find the LCM of the indices, convert each radical to the common index, then multiply.
Can you multiply radicals with different indices?
Yes, multiply radicals with different indices by converting both to a common index (the LCM of the original indices). For example, sqrt2 x ^3sqrt4 = 2^(1/2) x 4^(1/3) = 2^(3/6) x 4^(2/6) = ^6sqrt(8x16) = ^6sqrt128 = 2·^6sqrt2. This calculator handles radicals with different indices automatically.
What is the product rule for radicals?
The product rule for radicals states that ⁿsqrta x ⁿsqrtb = ⁿsqrt(axb), provided the radicals are real numbers and the same index is used. This rule allows combining two radicals into one by multiplying the radicands while keeping the index unchanged.
How do you simplify radical products?
After multiplying the radicands, simplify the resulting radical by extracting perfect nth powers. For example, sqrt8 x sqrt2 = sqrt16 = 4, and 2sqrt12 x 3sqrt27 = 6sqrt324 = 6x18 = 108. This calculator automatically simplifies the product to its simplest radical form.
What if the radicand is negative?
For even indices (square root, fourth root, etc.), radicands must be non-negative to produce real results. For odd indices (cube root, fifth root, etc.), negative radicands are allowed. This calculator works with non-negative radicands for all cases.
How is multiplying radicals used in real life?
Multiplying radicals appears in geometry (areas and volumes of irregular shapes), physics (wave interference, root-mean-square calculations), engineering (structural analysis with square-root relationships), statistics (standard deviation computations), and computer graphics (distance calculations in 3D space).
What is the difference between multiplying radicals and multiplying exponents?
Radicals are a specific type of exponent (fractional exponents). Multiplying radicals uses the rule ⁿsqrta x ⁿsqrtb = ⁿsqrt(ab), while multiplying exponents uses a^m x a^n = a^(m+n). Radicals are essentially exponents in fractional form, so the underlying principles are related but applied differently.