Question

3. Using prime factorisation, determine which am
the following are perfect squares:
b. 1009
​

Answer 1

Answer:

$A perfect square can always be expressed as a product of equal factors.</p><p></p><p>(i)</p><p>Resolving into prime factors:</p><p> 441=49×9=7×7×3×3=7×3×7×3=21×21=(21)2441=49×9=7×7×3×3=7×3×7×3=21×21=(21)2</p><p></p><p>Thus, 441 is a perfect square.</p><p></p><p>(ii)</p><p>Resolving into prime factors:</p><p> 576=64×9=8×8×3×3=2×2×2×2×2×2×3×3=24×24=(24)2576=64×9=8×8×3×3=2×2×2×2×2×2×3×3=24×24=(24)2</p><p></p><p>Thus, 576 is a perfect square.</p><p></p><p>(iii)</p><p>Resolving into prime factors: </p><p>11025=441×25=49×9×5×5=7×7×3×3×5×5=7×5×3×7×5×3=105×105=(105)211025=441×25=49×9×5×5=7×7×3×3×5×5=7×5×3×7×5×3=105×105=(105)2 </p><p></p><p>Thus, 11025 is a perfect square.</p><p></p><p>(iv)</p><p>Resolving into prime factors:</p><p>1176=7×168=7×21×8=7×7×3×2×2×21176=7×168=7×21×8=7×7×3×2×2×2</p><p></p><p>1176 cannot be expressed as a product of two equal numbers. Thus, 1176 is not a perfect square.</p><p></p><p>(v)</p><p>Resolving into prime factors:</p><p> 5625=225×25=9×25×25=3×3×5×5×5×5=3×5×5×3×5×5=75×75=(75)25625=225×25=9×25×25=3×3×5×5×5×5=3×5×5×3×5×5=75×75=(75)2</p><p></p><p> Thus, 5625 is a perfect square.</p><p></p><p>(vi)</p><p>Resolving into prime factors:</p><p> 9075=25×363=5×5×3×11×11=55×55×39075=25×363=5×5×3×11×11=55×55×3</p><p></p><p>9075 is not a product of two equal numbers. Thus, 9075 is not a perfect square.</p><p></p><p>(vii)</p><p>Resolving into prime factors:</p><p> 4225=25×169=5×5×13×13=5×13×5×13=65×65=(65)24225=25×169=5×5×13×13=5×13×5×13=65×65=(65)2 </p><p></p><p>Thus, 4225 is a perfect square.</p><p></p><p>(viii)</p><p>Resolving into prime factors: </p><p>1089=9×121=3×3×11×11=3×11×3×11=33×33=(33)21089=9×121=3×3×11×11=3×11×3×11=33×33=(33)2 </p><p></p><p>Thus, 1089 is a perfect square.$